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The present invention relates generally to digital copy protection, digital rights management, and conditional access, and more particularly but not exclusively to enabling transferable entitlements using Entitlement Management Messages (EMMs) for providing content to different network devices.
Today a consumer can readily purchase an entitlement to content such as a ticket to the opera, a sports event, movie, or the like. Often, the purchased ticket can be redeemed at some later stage and location. Similarly a consumer may purchase an airline ticket and redeem it for an airplane flight. However, there is a difference of transferability between these two ticket transactions. For various reasons, of both pricing and security, airline tickets represent non-transferable entitlements, where only the named recipient of the entitlement may redeem it, whereas movie tickets, or the like, are typically transferable.
Transferability is an attribute of the entitlement granted by an original owner to the recipient. It means that the recipient may be free to resell or transfer title to the entitlement prior to its redemption. It also typically means that the owner or its distributors agree to honor the redemption of the entitlement from whoever presents the entitlement. Thus, in some situations, a transferable entitlement may become an object of trade.
However, in today's realm of content, such as in the Internet Protocol Television (IPTV) domain, or the like, entitlements do not readily support transferability. If a recipient were to purchase an entitlement on one set top box (STB) there presently is no mechanism to enable the transfer of that entitlement to another set top box or other network device for redemption. Transfer of entitlements between devices on the same or different networks may open a wealth of opportunity for consumers and for content providers.
Moreover, IPTV, and the like, may be currently served in discrete networks—so-called ‘walled-garden’ networks. These networks typically ensure a level of quality of service and security. However the walls often impose a barrier to a market of consumers inside the wall. The broader commercial motivation of this invention therefore includes allowing third-party content providers outside the walls to gain access to this market. Thus, it is with respect to these considerations and others that the present invention has been made |
Mind The Gap
America’s British population has taken to the web to voice its displeasure at news that U.S. candy giant Hershey has successfully blocked our much loved U.K.-produced chocolate from being exported to the land of the free.
So the Oscar nominations were announced this morning, and, as expected, the great British hope, Atonement, was nominated for Best Picture. However, its two stars, James McAvoy and Keira Knightley, were omitted for the top acting …
The British dominance of Hollywood has been a big story throughout award season. But one could be forgiven for mistaking last night’s Oscars for a World Cup match, and, predictably, Britain got beaten by Mexico.
Latest Interviews
The Latest from Mind The Gap
America’s British population has taken to the web to voice its displeasure at news that U.S. candy giant Hershey has successfully blocked our much loved U.K.-produced chocolate from being exported to the land of the free. |
This invention relates to a method for applying normally dry relatively large particle size (granular) fertilizers to crops, such as lawns. Lawn fertilizers are available in various forms including solutions of nutrients in water, dispersions (suspensions) of fine powders (70-80 mesh and smaller) in an aqueous medium, dry powders and dry granules. In some cases, the nutrient materials are supported on an inert carrier, e.g. sand or clay.
Both liquid fertilizers and dispersions of fine powders in aqueous mediums are usually spray applied using conventional types of liquid solution fertilizer spraying equipment. A typical example of a spray applied dispersion of a powdered fertilizer material is illustrated by the U.S. Pat. No. to Funk 4,036,627. This patent discloses a high analysis fertilizer formulaton of low bulk density powdered ureaformaldehyde having soluble and insoluble portions combined with soluble monopotassium phosphate in which the resultant mixture is a dry homogeneous blend, free of fillers and binding agents, and which may be carried in a liquid medium for application to surface or subsurface areas by conventional liquid solution fertilizer applying equipment. The suspension generally has a fairly high concentration of the fine powder particles in the liquid medium.
Dry fertilizers in the powder form or the granular form are conventionally applied by dry spreaders. Numerous examples of dry powdered and granular fertilizer compositions are well known to those skilled in the art. Recently, these have begun to be formulated with provisions for timed (slow) release of the nutrients to avoid "burning" the crop and to reduce the number of applications in a growing season.
Each of the various physical forms of fertilizer compositions has its advantages and disadvantages. Spray applied liquid fertilizer solutions and dispersions of powdered nutrient materials are characterized by the ability to be applied evenly and from a tank truck, for example. These fertilizer forms usually provide nutrients which are immediately available to the lawn, and therefore enable quick response of the lawn to the application, i.e. quick "greening" of the lawn. However, such liquid solutions are often too rich in immediately available nutrients, particularly nitrogen. A solution which is too rich in nutrients can cause "burning" of the lawn. Additionally, insect and fungus growth may be accelerated. Still further, liquid solution type fertilizers do not often possess long life on or in the ground and their effect is quickly lost. Frequent application is required to maintain a desired nutrient level in the soil during a growing season.
With the finely divided powder or dispersion, a principal problem is retention on the leaves or blades of grass. This can also cause burning. Additionally, ambient conditions and normal lawn care procedures may result in loss of a significant value of the fertilizer. For example, application of dry powder is usually accompanied by considerable dusting and wind loss. Moreover, when the lawn is cut, and the clippings collected, a substantial portion of a powdered fertilizer, whether dry or dispersion applied, is carried away and lost. With a rotary lawn mower, dusting of a powdered fertilizer can also be a problem.
Granular fertilizers which are spread on the lawn in a dry condition, do not generally have the foregoing types of application problems encountered with powdered fertilizers. Because of the larger particle size, dusting is not a problem. Further, retention on the blades of grass or on leaves is not generally a problem with granular fertilizers. Thus, loss on removal of grass clippings is negligible. However, like any spreader applied fertilizer, application is usually uneven because of turns at the end of a row, skips, overlaps, etc. Without care, overfertilizing can occur in certain areas and under fertilizing in others. A blotchy appearance results. Furthermore, the immediate nutrient availability of granular fertilizers may be lost due to leaching. Thus, with granular fertilizers obtaining quick "greening" can be a problem. Thus, as can be seen from the foregoing discussion the problems which are often encountered in the application of liquid, liquid dispersion or dry spread granular fertilizers are also manifested in the quality of performance of the fertilizer. |
Recently, I sat down to talk with a group of eight students from a large prominent church in Southern California. They were raised in the church. They were regulars at youth group. They claimed to be in relationship with Christ. Yet, they were dead. As I tried to engage them, most seemed unmoved and uninterested. And I was not surprised.
As I work with churches around the country, I encounter countless Christian students who are apathetic toward spiritual things. Their relationship with Christ is passionless. Talk of God is ho-hum. But why? Shouldn’t our relationship with Christ be life’s most exciting adventure? I’m not suggesting the Christian life is one, big, emotional high, but why are students more willing to plug into their iPods than their Bibles? Why are they more excited about the latest celebrity gossip than the Gospel? Why aren’t their lives filled with the drama of God’s Kingdom?
I think a big part of the problem is that Christian students rarely engage their world for the cause of Christ. Here’s what I’ve observed in my training over the years. The most exciting events I do, the events where students seem to come to life, are those where there is some component of engagement. Let me illustrate.
For almost ten years now, I’ve been taking students on mission trips to Berkeley and Utah. Each trip requires hours of training, typically in the form of classroom instruction and the reading of required books. This training is important and necessary, but it’s not what generates the most buzz among the students. Students get fired up on the trip when we give them opportunities to engage non-believers.
On these trips we invite Mormon leaders, Unitarians, gay activists, Hare Krishna priests, skeptics, and atheists to dialogue with students. We give our non-Christian guests time to share their views, followed by a time of questions from our students. It’s during Q&A when students really come to life. They ask question after question, graciously yet firmly force our skeptical guests to give a reason for their views. At the conclusion of each encounter, we thank our guests and then spend time debriefing. At this point, students are always abuzz, asking me question after question. Before I know it, an hour of discussing apologetics and theology with youth will have flown by.
In addition, we send our groups onto college campuses, like BYU or Berkeley, to conduct surveys. The surveys are designed to get our students into conversation with non-Christians students about spiritual issues. At first, students are fearful and anxious. They’re skeptical about people’s willingness to engage with them. But after an hour or two of surveys, students return and they are always pumped. During our debrief time, students can’t wait to share about their encounters. They’re filled with excitement about their conversations on campus with non-Christians.
When we create opportunities for students to engage, there is a vibrancy that infuses the events. But this shouldn’t come as a surprise. Christianity is not a spectator sport. Our teaching should not remain in a classroom or behind the four walls of the church. If we want to train students who can defend the faith not just intelligently but passionately, we need to get them in the game. Think about any sports teams. It’s the starters who are the most passionate about the game, right? The benchwarmers, not so much. I think that’s one reason why our mission trips to Berkeley and Utah are exciting and successful. They get students in the game. They get students engaging a lost world with the truth of Jesus Christ.
In 2014, students will get a taste of being in the game as I take them to Berkeley and Utah. I’ve already maxed out the number of mission trips I’m capable of taking through July. Indeed, we’ve had to turn groups away or ask them to start scheduling for 2015. So this year, we’ll be getting students off the sidelines and igniting their fire for Christ. I can’t wait.
As a parent of 5 kids, summer gets expensive. I have to pay for swim lessons, soccer camps, VBS, youth group trips, family vacations, and more. And these costs don't even include feeding my kids all … > Read full article |
Gerda Gilboe
Gerda Gilboe (5 July 1914 – 11 April 2009) was a Danish actress and singer. She appeared in 18 films between 1943 and 2003.
Life
Gilboe was born in 1914. She was the daughter of a blacksmith, Gilboe started her career in musical theatre and operas in Aarhus before she moved to Copenhagen to work at different theatres. Her national breakthrough came, when she accepted the role as Eliza in My Fair Lady at Falkoner Teatret at short notice in 1960. Although she was then in her mid-40s and had only five days to learn the part, the production was a huge success.
In the following years she took on more and more non-singing roles, and besides her theatre career she took a degree in rhetoric. Later in her life she started teaching rhetoric and drama. She appeared in several films, receiving particular acclaim for her appearance as Esther in Carlo & Esther, a 1994 film. She plays a woman in her 70s who catches the attention of Carlo who has a wife with Alzheimer's disease. Rides on his motorbike lead to an affair.
Death
Gilboe died on 11 April 2009 at an actors' home in Copenhagen, aged 94.
Filmography
A Time for Anna (2003)
Kærlighed ved første hik (1999)
Dybt vand (1999)
Besat (1999)
Antenneforeningen (1999)
Kun en pige (1995)
Elsker elsker ikke... (1995)
Carlo & Ester (1994)
Lad isbjørnene danse (1990)
Isolde (1989)
Sidste akt (1987)
Walter og Carlo – yes, det er far (1986)
Pas på ryggen, professor (1977)
Kun sandheden (1975)
Den kyske levemand (1974)
Lise kommer til Byen (1947)
En ny dag gryer (1945)
Moster fra Mols (1943)
References
External links
Category:1914 births
Category:2009 deaths
Category:Danish female singers
Category:Danish film actresses
Category:Danish musical theatre actresses
Category:People from Aarhus
Category:Place of birth missing
Category:Place of death missing
Category:20th-century Danish actresses
Category:20th-century singers
Category:20th-century women singers |
Abstract
The entorhinal cortex receives a large projection from the piriform cortex, and synaptic plasticity in this pathway may affect olfactory processing. In vitro whole cell recordings have been used here to investigate postsynaptic signalling mechanisms that mediate the induction of long-term synaptic depression (LTD) in layer II entorhinal cortex cells. To induce LTD, pairs of pulses, using a 30-millisecond interval, were delivered at 1 Hz for 15 minutes. Induction of LTD was blocked by the NMDA receptor antagonist APV and by the calcium chelator BAPTA, consistent with a requirement for calcium influx via NMDA receptors. Induction of LTD was blocked when the FK506 was included in the intracellular solution to block the phosphatase calcineurin. Okadaic acid, which blocks activation of protein phosphatases 1 and 2a, also prevented LTD. Activation of protein phosphatases following calcium influx therefore contributes to induction of LTD in layer II of the entorhinal cortex.
1. Introduction
The mechanisms that mediate the
induction of long-term synaptic potentiation (LTP) [1, 2] and depression (LTD) [3–5]
have been studied intensively within the hippocampus, but less is known about the
signalling mechanisms for LTP and LTD in the entorhinal cortex. Because the
entorhinal cortex receives highly processed inputs from sensory and association
cortices and also provides the hippocampal region with much of its sensory
input [6, 7], lasting changes in the strength of synaptic inputs to the entorhinal
cortex could alter the manner in which multimodal cortical inputs are
integrated, modulate the strength of transmission of specific patterns of sensory
input within the hippocampal formation, and contribute to mnemonic function [8–11]. Determining the effective stimulation
parameters and the intracellular signals that mediate synaptic plasticity in
the entorhinal cortex should allow insight into basic mechanisms that contribute
to the cognitive functions of the parahippocampal region.
Long-term potentiation of cortical inputs to
the superficial layers of the entorhinal cortex has been described in vivo [11–14] and in vitro [15, 16]. Stimulation
patterns required to induce LTP tend to be more intense in the entorhinal
cortex than in the hippocampus [12, 14], and we have also found that induction
of LTD in the entorhinal cortex requires intense low-frequency stimulation [17, 18]. In the hippocampus, conventional
1 Hz stimulation trains have been most
effective in slices taken from juvenile animals [19, 20] but are generally
ineffective in adult slices [21–23] and in intact animals ([31, 32], see also [33]).
Similarly, 1 Hz stimulation induces entorhinal LTD in slices from young animals
[28, 29] but is not effective in vivo [17] or in slices from older animals [18].
Repeated stimulation using pairs of pulses separated by a short 25- to 50-millisecond
interval can induce LTD more effectively in both the CA1 ([24–26], but see [27]) and
entorhinal cortex [17, 18, 33, 34]. In the CA1, the LTD induced by this stimulation
pattern is NMDA receptor-dependent, but it also depends upon activation of
local inhibitory mechanisms by the pulse-pairs [30, 31]. In the entorhinal cortex, however, repeated
paired-pulse stimulation using a 10-millisecond interval that evokes maximal paired-pulse
inhibition does not induce LTD, and LTD is induced when a 30-millisecond
interval is used that evokes maximal paired-pulse facilitation [17]. The LTD can also be enhanced when GABAA transmission is reduced with bicuculline [18]. This further suggests that LTD
in the entorhinal cortex does not require activation of local inhibitory
mechanisms but rather requires prolonged stimulation patterns that are strong
enough to overcome local inhibition and lead to NMDA receptor activation. Strong
local inhibition in the entorhinal cortex [8, 35] may thus place a restraint on
activity-dependent synaptic modification. Consistent with this idea is the
finding that the same pairing stimulation protocol that induces LTP in
hippocampus leads to LTD in entorhinal cortex [28].
Signalling mechanisms that mediate LTD in the
superficial layers of the entorhinal cortex share some similarities with NMDA
receptor-dependent LTD in the hippocampus. Long-term depression of superficial
layer inputs to layer II is dependent on NMDA receptor activation both in vivo
and in vitro [17, 18, 28, 33] but does not require activation of group I/II
metabotropic glutamate receptors ([18, 28], see [36, 37]). In the hippocampus, moderate
and prolonged influx of calcium via NMDA receptors activates calmodulin which leads
to LTD via activation of the protein phosphatase calcineurin (PP2b). Calcineurin
increases the activity of protein phosphatase 1 by reducing the activity of
inhibitor 1, and this can cause rapid reductions in AMPA-mediated responses [2, 38, 39]. Hippocampal LTD is expressed partly through the reduced conductance of
AMPA receptors caused by dephosphorylation of the GluR1 subunit by PP1 [2, 4], but
careful study has shown that calcineurin-dependent LTD in deep layer inputs to
layer II neurons in the young entorhinal cortex is not associated with a
reduced AMPA conductance, but rather involves internalization of AMPA receptors
and their proteosome-mediated degradation [28].
In the present study, the early postsynaptic
signalling mechanisms that mediate LTD in layer I inputs to layer II neurons of
the medial entorhinal cortex have been investigated using recordings of whole
cell excitatory postsynaptic potentials. Long-term depression was induced using
a prolonged paired-pulse stimulation pattern that was previously found to be
effective for induction of NMDA-receptor-dependent LTD [18]. Pharmacological agents applied to the
bathing medium or intracellular solution were used to assess the dependence of
LTD on calcium-dependent signalling mechanisms including the phosphatases calcineurin
and PP1/PP2a.
2. Experimental Procedures
2.1. Slices and Whole Cell Recordings
Experiments were performed on slices from male
Long-Evans rats (4 to 8 weeks old). Animals were anesthetized with halothane and brains were rapidly removed
and cooled (4°C) in oxygenated artificial cerebrospinal fluid (ACSF). ACSF
consisted of (in mM) 124 NaCl, 5 KCl, 1.25 NaH2PO4, 2
MgSO4, 2 CaCl2, 26 NaHCO3, and 10 dextrose and
was saturated with 95% O2–5% CO2. All chemicals were obtained
from Sigma (St. Louis, Mo, USA) unless otherwise indicated. Horizontal slices (300𝜇m) were cut with a
vibratome (WPI, Vibroslice
NVSL, Sarasota, Fla, USA) and were allowed to recover for at least one hour before
recordings. Slices were maintained in a recording chamber with oxygenated ACSF
at a rate of 2.0 mL/min, and a temperature from 22 to 24°C was used to
minimize metabolic demands on slices [18, 28]. Neurons were viewed with an
upright microscope (Leica
DML-FS, Wetzlar, Germany) equipped with a 40x objective, differential interference
contrast optics, and an infrared video camera (Cohu, 4990 series, San Diego, Calif, USA).
2.2. LTD Induction and Pharmacology
Whole-cell current clamp recordings of EPSPs
were monitored 10 minutes before and 30 minutes after LTD induction by
delivering test-pulses every 20 seconds. Intensity was adjusted to evoke EPSPs
that were approximately 3 to 4 mV in amplitude, and cells were held 5 mV below
threshold when necessary to prevent the occurrence of spikes in response to
EPSPs. Stimulus parameters for LTD
induction were based on those used previously in vivo and in vitro
[17, 18]. The induction of LTD was tested using pairs of stimulation pulses (30-millisecond
interpulse interval) delivered at a frequency of 1 Hz for either 7.5 or 15 minutes
[18]. Control cells received test-pulses throughout the recording period and
did not receive conditioning stimulation.
Signalling mechanisms mediating the induction
of LTD were tested using stock solutions of pharmacological agents that were
stored frozen and diluted on the day of use. NMDA glutamate receptors were
blocked by constant bath application of 50𝜇M DL-2-amino-5-phosphonovalerate
(APV). The calcium chelator 1,2-bis(2-aminophenoxy)-ethane-N,N,N′N′-tetraacetic
acid (BAPTA, 10 mM) was included in the recording electrode solution to block
increases in intracellular calcium. To block activation of the
calmodulin-dependent protein phosphatase calcineurin (PP2b) slices were pre-exposed
to 250𝜇M cyclosporin A (Toronto Research Chemicals Inc., North York, Ontario, Canada) for 1.5 to 3 hours [39]. In other
experiments, FK506 (50𝜇M) was included in the recording electrode solution to
block calcineurin [39, 40]. In other experiments, okadaic acid (0.1 or 1.0𝜇M)
was included in the recording solution to block activation of protein phosphatases
1 and 2a [40, 41]. Control recordings without paired-pulse stimulation were
used to verify the stability of recordings in cells filled with FK506 and 1.0𝜇M
okadaic acid.
2.3. Data Analysis
Synaptic
responses and electrophysiological properties of layer II neurons were analyzed
using the program Clampfit 8.2 (Axon Instr.). Data were standardized to the mean of baseline responses for plotting
and were expressed as the mean ±SEM. Changes in EPSP amplitude were assessed
using mixed-design ANOVAs and Neuman-Keuls tests that compared the average responses
during the baseline period, 5 minutes after conditioning stimulation, and during
the last 5 minutes of the recording period.
Layer II neurons
were classified as putative stellate
or nonstellate neurons based on electrophysiological characteristics described
by Alonso and Klink [42]. Stellate neurons were characterized by the presence
of low-frequency subthreshold membrane potential oscillations, a depolarizing
afterpotential following spikes, and prominent inward rectification in response
to hyperpolarizing current pulses. Both
pyramidal and stellate neurons in layer II can show inward rectifying sag
responses [43]. Here, neurons recorded were clearly in layer II, usually near
the border with layer I, and a proportion of these neurons did not show clear
sag and were classified as pyramidal neurons. Input resistance was
determined from the peak voltage response to −100 pA current pulses (500-millisecond
duration), and rectification ratio was quantified by expressing peak input
resistance as a proportion of the steady-state resistance at the end of the current
pulse.
3. Results
Stable recordings were obtained from 57 putative
stellate neurons and 21 putative nonstellate cells. Peak input resistance was
similar in stellate and pyramidal neurons (stellate, 95 ± 6 MΩ; pyramidal, 96
± 10 MΩ) but there was a much larger sag in voltage responses to hyperpolarizing
current injection in stellate cells (rectification ratio 1.37±0.04 in stellate
cells versus 1.06±0.01 in pyramidal cells). The amplitude of baseline synaptic
responses evoked by layer I stimulation was similar in stellate (3.9±0.2 mV)
and pyramidal cells (3.7±0.4 mV), and the amount of depression induced was
also similar for recording conditions in which significant LTD was obtained (71.2±5.6% in 14 stellate and 76.8±7.6% in 6 pyramidal cells).
3.1. LTD Induction
To determine if a relatively brief LTD
induction protocol could be used to induce LTD in whole-cell recordings, the
first tests attempted to induce LTD using paired-pulse delivery at 1 Hz for 7.5
minutes (𝑛=10) which can induce
moderate LTD of field potentials in a gas-fluid interface recording chamber [18].
Paired-pulse stimulation for 7.5 minutes did not induce depression of EPSPs
relative to control cells (93.0±10.0% of baseline after 30 minutes; F2,28=0.09,𝑃=.92). We previously observed stronger LTD of field potentials in the
interface recording chamber after 15 minutes versus 7.5 minutes of paired-pulse
stimulation [18], and prolonged
paired-pulse stimulation for 15 minutes also reliably induced LTD of whole-cell
EPSPs (𝑛=7, Figure 1). EPSP amplitude was reduced to 56.3±9.5% of
baseline levels 5 minutes after the conditioning stimulation, and remained at
58.6±6.1% of baseline levels at the end of the 30 minutes follow-up
period (F2,22=14.2,𝑃<.001).
Responses in control cells were stable (𝑛=6),
and remained at 99.6±2.6% of baseline levels at the end of the recording period (Figures 1(b2), 1(c)).
Figure 1: Prolonged, low-frequency stimulation induces long-term depression of
EPSPs in neurons in layer II of the entorhinal cortex. (a) The location of stimulating and recording
electrodes in acute slices containing the entorhinal cortex. (b) and (c) Long-term
depression was induced by repetitive delivery of pairs of stimulation pulses at a rate of 1 Hz for 15 minutes (PP-LFS). The amplitude
of synaptic responses remained stable in control cells that did not receive conditioning
stimulation. Traces in (b) compare responses
recorded during the baseline period (1) and during the follow-up period (2) in a
neuron that received low-frequency stimulation (b1) and in a control cell (b2). Responses were obtained at the times indicated in (c).
Averaged points in (b) indicate the mean ±1 SEM in this
and subsequent figures. (d) Long-term depression was not
reliably induced when low-frequency stimulation was delivered for only 7.5 minutes
rather than 15 minutes, indicating that induction of LTD requires prolonged
stimulation.
3.2. NMDA Receptors and Postsynaptic Calcium
The NMDA receptor antagonist MK-801 blocks
induction of LTD in the entorhinal cortex in vivo [17] and the NMDA receptor blocker APV has been shown to prevent LTD
of field potentials and EPSPs in entorhinal cortex slices [18, 28, 33]. We therefore tested for
the NMDA receptor-dependence of LTD of EPSPs in the current preparation using
constant bath application of APV (50𝜇M). Induction of LTD by 15 minutes of
paired-pulse stimulation was blocked by APV (𝑛=6, Figure 2(a)). There was a tendency for responses to be
potentiated immediately following conditioning stimulation, but this variable effect
was not statistically significant, and responses were close to baseline levels
at the end of the recording period (96.7±13.2% of baseline;
F2,10=2.99,𝑃=.09).
Figure 2: The induction of long-term
depression is dependent on activation of NMDA glutamate receptors and on
increases in postsynaptic calcium. (a) Constant bath application of the
NMDA receptor antagonist APV (50𝜇M) blocked the induction of long-term
depression by 15 minutes of paired-pulse low-frequency stimulation (PP LFS). (b) Blocking increases in
postsynaptic calcium by including the calcium chelator BAPTA (10 mM) in the recording
electrode solution also blocked the induction of LTD. The transient facilitation of EPSPs immediately
following stimulation was significant for the BAPTA condition but not the APV
condition, and responses were at baseline levels at the end of the recording
periods. The block of lasting depression suggests that calcium influx via NMDA
receptors is required for induction of LTD.
The role of postsynaptic calcium in LTD
induction was tested by recording from cells in which the calcium chelator
BAPTA (10 mM) was included in the recording electrode solution (10 mM, 𝑛=6, Figure 2(b)). Cells filled with
BAPTA had longer-duration action potentials than control cells (6.1±0.7 versus
3.3±0.1 milliseconds measured at the base; 𝑡1,9=3,57,𝑃<.01) consistent with a reduction in
calcium-dependent potassium conductances. The induction of LTD was blocked in
cells loaded with BAPTA. There was a significant increase in the amplitude of
EPSPs immediately following paired-pulse stimulation (to 122.3±6.0% of baseline; F2,10=5.46,𝑃<.05; N–K, 𝑃<.05), but responses returned to
baseline levels within 10 minutes and were at 94.8±7.1% of baseline levels
after 30 minutes (N–K, 𝑃=0.50, Figure
2(b)). An increase in postsynaptic calcium is therefore required for induction
of LTD in layer II neurons of the entorhinal cortex.
3.3. Protein Phosphatases
The role of the calmodulin-dependent protein phosphatase
calcineurin (PP2b) in LTD in layer II neurons was tested using either
pre-exposure to 250𝜇M cyclosporin A in the bathing medium [39], or by
including 50𝜇M FK506 postsynaptically in the recording electrode solution. In cells pre-exposed to cyclosporin A, paired-pulse stimulation was followed by a depression in EPSP amplitude that
reached
82.4±7.5% of baseline levels after 30 minutes (Figure 3(a)).
Although the depression in the cyclosporin group was not statistically
significant (F2,10=3.51,𝑃=0.07,𝑛=6), the depression obtained was also not significantly less than that
observed in control ACSF (F1,11=3.79,𝑃=.08). The result was therefore ambiguous with respect to the role
of calcineurin in LTD. To test the involvement of calcineurin more definitively
and to avoid potential presynaptic effects, the calcineurin blocker FK506 was
included in the recording electrode solution for additional groups of cells [40].
Responses in cells filled with FK506 showed a significant potentiation
immediately following paired-pulse stimulation (𝑛=8), but there was no lasting
change in response amplitudes in comparison to control cells filled with FK506
that did not receive conditioning stimulation (𝑛=7). Responses were increased
to 134.9±10.5% of baseline levels immediately following paired-pulse
stimulation, (F2,26=7.71,𝑃<.01; N–K, 𝑃<.001;𝑛=8) but returned to 102.2±6.1% of baseline levels after 30 minutes (Figure 3(b)).
Figure 3: Long-term depression is dependent
on activation of the calmodulin-dependent protein phosphatase calcineurin.
Although LTD was only partially inhibited by pre-exposure to cyclosporin A, it
was completely blocked when FK506 was included in the recording electrode
solution. (a) Pre-exposure of slices to the calcineurin inhibitor cyclosporin
A (250𝜇M) for 1.5 to 3 hours resulted in a partial block of LTD by repeated
paired-pulse stimulation. The amount of LTD induced was smaller than in control
ACSF and was close to statistical significance (𝑛=6,𝑃=.07). (b) Including the FK506 in the recording electrode solution to directly block
postsynaptic calcineurin prevented the induction of LTD. Analysis of group responses showed a
significant increase in responses during the baseline period, but responses in
control cells indicate that this increase is transient and unlikely to have
affected measurement of LTD. Inhibition of postsynaptic calcineurin therefore
prevents induction of LTD in layer II cells of the entorhinal cortex.
Inspection of averaged responses suggested that
there was an initial increase in responses during the baseline period among
cells filled with FK506, and comparison of responses recorded during the first
and last minutes of the baseline period showed that the increase was
significant (𝑡14=3.09,𝑃<.01).
Interestingly, then, interfering with calcineurin function can lead to enhanced
basal synaptic transmission in entorhinal neurons. This increase is not likely
to have affected measures of LTD in conditioned cells, however, because control
responses showed only a transient increase after which responses remained stable.
Protein phosphatase 1 is thought to contribute directly
to suppression of hippocampal EPSPs during LTD by dephosphorylation of the GluR1
AMPA receptor subunit. The involvement of PP1 to LTD in the entorhinal cortex
was therefore tested by including okadaic acid in the recording electrode
solution. In early experiments, a low concentration of 0.1𝜇M okadaic acid [41] did not block LTD
induction, and responses were depressed to 72.7±8.7% of baseline levels at the
end of the recording period (F2,24=4.65,𝑃<.05; N–K, 𝑃<.001;𝑛=8). However, increasing the concentration of okadaic
acid to 1.0𝜇M [40] blocked the
induction of LTD. There was a variable and nonsignificant reduction in
responses immediately following conditioning stimulation (to 89.0±14.9% of
baseline) and responses were also near baseline levels after 30 minutes (96.0±6.6% of baseline 30; F2,22=0.18,𝑃=.84;𝑛=7; Figure 4). Activation of PP1 is therefore likely to contribute
to mechanisms of LTD in the entorhinal cortex.
Figure 4: The induction of LTD was blocked in
a dose-dependent manner by including okadaic acid in the recording electrode
solution to block activation of protein phosphatase 1 (PP1). (a)
and (b) A low concentration of 0.1𝜇M okadaic acid failed to block LTD
induction, but raising the concentration to 1.0𝜇M resulted in a block of LTD
induction (compare traces in A1 versus A2). Responses in
control cells filled with 1.0𝜇M okadaic acid that did not receive conditioning
stimulation remained stable. The block of LTD by okadaic acid suggests that activation
of PP1 mediates LTD in the entorhinal cortex.
4. Discussion
The current paper has used prolonged repetitive
paired-pulse stimulation to induce LTD in layer I inputs to layer II neurons of
the medial entorhinal cortex and has determined the early postsynaptic signals
that mediate LTD in these cells. Consistent with previous observations, the LTD
observed here was obtained in both putatively identified stellate [28] and
pyramidal [44] cells. The induction of LTD was blocked by the NMDA glutamate
receptor antagonist APV, and by the calcium chelator BAPTA, indicating that
calcium influx via NMDA receptors is required for LTD. The induction of LTD was
also blocked by the calcineurin inhibitor FK506, and by okadaic acid which
blocks activation of protein phosphatases 1 and 2a. Calcineurin is required for
LTD of deep layer inputs to layer II stellate cells [28], and
calcineurin-dependent activation of PP1 contributes to NMDA receptor-dependent
LTD of AMPA responses in the hippocampus [2, 4].
The dependence of LTD in the entorhinal cortex
on activation of NMDA receptors has been a consistent finding in vivo and in
slices. It has been observed following stimulation protocols including 1 Hz trains,
pairing of presynaptic stimulation at 0.33 Hz with postsynaptic depolarization
[28], repeated paired-pulse stimulation [18, 33], and spike-timing-dependent
induction of LTD [44]. Long-term depression was blocked by including the calcium
chelator BAPTA in the recording electrode solution (Figure 2) [28], and this is
consistent with calcium influx via NMDA receptors as a critical trigger for
entorhinal LTD. Metabotropic glutamate receptor activation and release of
calcium from intracellular stores can contribute to LTD in the hippocampus [2, 36, 37, 45], but activation of metabotropic glutamate receptors is not required
for entorhinal LTD [18, 28]. Calcium influx through voltage-gated calcium
channels can contribute to spike-timing-dependent LTD in the entorhinal cortex,
however. Cells with broadened action potentials that result in larger calcium transients show greater NMDA
receptor-dependent spike-timing-dependent LTD in layer II-III cells [44]. Calcium influx through voltage-gated
channels also mediates bidirectional spike-timing-dependent plasticity of
inhibitory synapses in entorhinal cortex [46]. A form of long-term depression
on layer V-VI neurons, expressed presynaptically through reduced transmitter
release, is also dependent on activation of voltage-dependent calcium channels
[33]. Calcium signalling mediated by voltage-gated channels therefore plays a
number of roles in modulating synaptic plasticity in the entorhinal cortex.
The contribution of the
calmodulin-dependent protein phosphatase calcineurin to LTD was tested by
incubating slices in cyclosporin A or by including FK506 in the recording
electrode solution. Cyclosporin A appeared to cause a partial block of LTD, and
responses were reduced to 82.4% of baseline as compared to 58.6%
in untreated cells (compare Figures 1(c)
and 3(a)), but the sizes of these LTD effects were not statistically different.
We obtained a more conclusive result with FK506, however, and LTD was
completely blocked by including FK506 in the recording electrode solution. Including
FK506 in the bathing medium has been used to block calcineurin-dependent
depression effects in entorhinal cortex [28], and in excitatory [47] and
inhibitory [48] synapses of the CA1 region. Here, we have loaded FK506 into the
recording electrode solution to avoid possible presynaptic effects of the drug
and to ensure that FK506 could act on calcineurin [39, 40, 49, 50]. The
block of LTD by FK506 indicates that LTD is dependent on calcineurin, and this
suggests that cyclosporin A resulted in only a partial block of calcineurin
activity.
Calcineurin is thought
to mediate expression of LTD in part by dephosphorylating inhibitor 1 and
thereby increasing the activity of PP1 [2, 4, 39]. The PP1/PP2a inhibitor okadaic
acid blocks LTD in the CA1 region [38, 40], and we have shown here that the induction
of LTD in the entorhinal cortex was blocked by including okadaic acid in the
recording electrode solution. This is the first report of LTD in the entorhinal
cortex dependent on PP1/PP2a. Protein phosphatases can regulate synaptic
function through a variety of mechanisms [51] that include dephosphorylation of
the ser-845 residue on the AMPA GluR1 subunit, and LTD in the entorhinal cortex
may be expressed partly through this mechanism. In addition, the work of Deng
and Lei [28] has found entorhinal LTD to be associated with a reduction in the
number of postsynaptic AMPA receptors, with no change in AMPA receptor
conductance, and has shown that this effect is dependent on proteosomes that degrade
AMPA receptors internalized through ubiquitinization. As in the hippocampus,
therefore, entorhinal LTD can be expressed through mechanisms involving
trafficking of AMPA receptors [52].
Long-term depression was induced here using
strong repetitive paired-pulse stimulation which we have used previously to
induce LTD in the entorhinal cortex in vivo and in slices ([17, 18], see also [33, 34]). LTD was induced following 15 minutes, but not 7.5 minutes of paired-pulse
stimulation; this is consistent with a requirement for prolonged activation of
calcium-dependent signalling mechanisms, and is also consistent with the
possibility that NMDA receptor-dependent metaplastic changes early in the train
may promote LTD induced by stimuli that occurred later in the 15-minute
duration trains [53]. We previously found 1 Hz stimulation to be ineffective in
vivo and in slices from Long-Evans rats [17, 18], but deep layer inputs to
stellate neurons in slices from 2 to 3 week-old Sprague-Dawley rats express
NMDA receptor-dependent LTD following 15 minutes of 1 Hz stimulation, or
following low-frequency stimulation paired with postsynaptic depolarization [28].
Thus, there may be developmental, strain-related, or pathway-specific factors
that affect the ability of 1 Hz stimulation to activate these signalling mechanisms.
The entorhinal cortex
is embedded within the temporal lobe through an extensive array of anatomical
connections [7] and has been linked behaviorally to a variety of sensory and
cognitive functions (e.g., [9, 10]). Lasting synaptic plasticity in the
entorhinal cortex is therefore likely to serve a variety of functions depending
on the synaptic pathways involved. Synaptic depression effects are generally thought
to complement synaptic potentiation during the formation of memory [45, 54–56], and it is possible that depression effects contribute to short and/or
long-term memory processing. However, the laminar architecture of the
entorhinal cortex, with superficial layers mediating much of the cortical input
to the hippocampal formation, suggests that long-term depression of synaptic
transmission in layer II may lead to long-term reductions in the salience of
particular elements or patterns of cortical input and may thus lead to lasting
changes in the multimodal inputs processed by the hippocampal formation. Similarly, the general resistance of the
entorhinal cortex to induction of LTD could serve to maintain relatively stable
information processing and integration of multimodal sensory inputs within the
medial entorhinal cortex.
Acknowledgments
This research was funded by grants to C. A.
Chapman from the Natural Sciences and Engineering Research Council of Canada
and the Canada Foundation for Innovation, and by a postdoctoral fellowship to
S.K. from Fondation Fyssen (France). C.A. Chapman is a
member of the Center for Studies in Behavioral Neurobiology funded by the Fonds
pour la Recherche en Santé du Québec.
A. Alonso, M. de Curtis, and R. Llinás, “Postsynaptic Hebbian and non-Hebbian long-term potentiation of synaptic efficacy in the entorhinal cortex in slices and in the isolated adult guinea pig brain,” Proceedings of the National Academy of Sciences of the United States of America, vol. 87, no. 23, pp. 9280–9284, 1990.View at Publisher · View at Google Scholar
S. M. Dudek and M. F. Bear, “Homosynaptic long-term depression in area CA1 of hippocampus and effects of N-methyl-D-aspartate receptor blockade,” Proceedings of the National Academy of Sciences of the United States of America, vol. 89, no. 10, pp. 4363–4367, 1992.View at Publisher · View at Google Scholar
M. F. Bear, “A synaptic basis for memory storage in the cerebral cortex,” Proceedings of the National Academy of Sciences of the United States of America, vol. 93, no. 24, pp. 13453–13459, 1996.View at Publisher · View at Google Scholar |
Divesting Of Kruger’s Cash (Updated)
Freshman Sen. David Carlucci, one of 17 Senate Democrats who received campaign contributions from Sen. Carl Kruger during the 2010 election cycle, is getting rid of that money after learning of the federal corruption charges lodged against his colleague earlier today.
“It is unfortunate that these types of allegations have clouded the legislature, tainting the hard working men and women who work diligently and honorably to serve their constituents,” Carlucci said in a statement.
“I ran on a platform of ethics reform and these unsavory allegations are just another example of why ethics reform in Albany needs to be addressed immediately. The people of New York deserve better. In light of these allegations, I will be donating the $5,000 given to me during my campaign from Senator Kruger to a charitable organization in my district.”
All told, Kruger, a prodigious fundraiser who had close to $1.9 million in his campaign committee, “Friends of Carl,” as of Jan. 15, doled out $49,000 to fellow senators this cycle, according to campaign finance records on file at the state Board of Elections. He also gave $450,000 to the DSCC.
Sen. Gustavo Rivera, who received $2,500 from Kruger, was the first lawmaker to announce he would divest himself of the scandal-scarred Brooklyn lawmaker’s contributions. Now, apparently, all four of the Independent Democratic Conference members – Carlucci, Diane Savino, Jeff Klein, and Dave Valesky – are following suit.
UPDATE: DSCC spokesman Josh Cherwin says:
“We will not be returning these funds, which were contributed during a previous election cycle and already spent.”
“This is yet another sad day for New York residents who rightfully expect integrity and accountability from their elected officials.”
“During the last election cycle, Senator Kruger’s campaign committee contributed a combined total of $8,000 to Independent Democratic Conference members. We decided to donate that amount to charitable organizations in our communities. We believe this to be the best use for this money.” |
Local graft-versus-host-reaction in mice specifically inhibited by anti-receptor antibodies.
The local graft-versus-host (GVH) reaction provoked by parental spleen cells in F1 mice was shown to be T-cell-dependent. GVH reactions were suppressed in F1 hybrid mice immunized with parental T lymphocytes of the same genotype, but not in F1 mice immunized with parental B cells. In some cases this immunity could be passively transferred by serum into normal F1 mice. The specific activity of such sera could be removed by absorption with either parental T or B cells. Some of the F1 antisera were specificlly cytotoxic for parental GVH-reactive lymphocytes. |
Holiday Punch — Plus a Cozy Fire
Charles Dickens gave us so much. Including this.
In A Christmas Carol,when Ebenezer Scrooge is presented with the Ghost of Christmas Present, he finds the "jolly Giant" sitting in state on an enormous heap of roast meats and other traditional English Christmas delicacies and flanked by "seething bowls of punch that made the chamber dim with their delicious steam."
Charles Dickens knew all about delicious steam.
He was a committed English traditionalist in his drinking. He didn't drink the international celebrity's customary champagne, champagne, and more champagne or the trendy drinks of his day — gin cocktails, claret cups, brandy smashes, or the like. Rather, his greatest affinity was for a drink that was fading faster and faster into the past by the time he came into fame. From 1700 to 1830, give or take a couple years on each end, the preeminent English social drink was the bowl of punch, a large-bore mixture of spirits (usually rum and cognac), citrus juice, sugar, water, and spice that was guaranteed to unite any gathering in jollity and boozy good cheer. But with the industrialization, commercialization, and urbanization of day-to-day life that the Victorian years brought, the convivial ritual of clustering around the flowing bowl became as quaint and outmoded as the tricorn hat.
Dickens, however, not only bucked the trend but made a whole performance out of bucking it. When he was among friends, it was his custom to brew up a bowl of punch, complete with a running disquisition on the techniques he was using and the ingredients he was deploying, thus adding instruction to delight (as one of his characters might say). Fortunately, in 1847, he wrote the whole procedure out for a friend's wife. It's not hard to follow, and there's no better way to get a holiday party started than by getting everybody involved in draining a bowl of punch. All it takes is a little preparation in advance, a willingness to hold forth a bit in front of your guests, and a high enough ceiling that you won't burn your house down. Dickens was never afraid to employ cheap sensationalism if it would help him get over, and there's nothing more sensational for selling a drink than setting it on fire. Here's our interpretation.
Advertisement - Continue Reading Below
Charles Dickens's Punch Ritual
For 12 to 16 people
Step 1: Three hours before your party, peel 3 lemons with a swivel-bladed vegetable peeler, trying to end up with three long spirals of peel. Put them in a 3- or 4-quart fireproof bowl with 3/4 cup demerara sugar or other raw sugar. Muddle the peels and the sugar together and let sit.
Talking Points: One of the secrets of punch making is to use the fragrant, sweet oil that resides in lemon peels as the sugar extract. The resulting sugar-oil mix ("oleo-saccharum") adds body to the punch.
Talking Points: The cognac is for body and smoothness, the strong rum for bouquet and (frankly) flammability, and the other rum for taming the strong one.
Step 3: Set 1 quart water to boil and put the bowl containing the lemon peels and sugar on a wooden cutting board or other heat-resistant surface in a spot where everyone can gather around. When the water boils, turn it off, gather your guests around the bowl, and pour in the cognac and rum, noting what you're adding and why.
Step 4: With a long-handled barspoon, remove a spoonful of the rum-cognac mixture and set it on fire. Return it to the bowl, using it to ignite the rest.
Stir with a ladle or long-handled spoon, attempting to dissolve the sugar. Let burn for 2 or 3 minutes, occasionally lifting one of the lemon peels up so people can admire the flames running down it.
Talking Points: You're setting the punch alight not because it looks cool but to burn off some of the more volatile elements of the alcohol. That's the story, anyway.
Step 5: Extinguish the flames by covering the bowl with a tray, and add the reserved lemon juice and the boiling water. (For cold punch, add 3 cups cold water, stir, and slide in a 1-quart block of ice, easily made by freezing a quart bowl of water for 24 hours.)
Step 6: Grate fresh nutmeg over the top and ladle out in 3-oz servings.
A Part of Hearst Digital Media
Esquire participates in various affiliate marketing programs, which means we may get paid commissions on editorially chosen products purchased through our links to retailer sites. |
Go back to /usr/src/vdr-1.2.5 and run runvdr.remote. If
you use Red Hat, set the environment variable
LD_ASSUME_KERNEL=2.4.1, because VDR doesn't yet work with the native
posix layer that Red Hat introduced in the latest version. The modules for
the DVB card then are loaded, and the VDR is started. Hook up your TV,
and you should see a black screen prompting you to define the keys on your
remote. After finishing the wizard, you're ready to watch TV, record
shows and remove commercials. You can listen to your MP3s and watch
videos. There's a manual in VDR's root that explains
how to record and edit TV events, using the time-shift feature.
Back It Up
In case you're disappointed that the end of the article is within reach,
don't worry; there still are some optional things you can do. The automatic
backup feature has some limitations. Although the (S)VCD backup works
flawlessly, the DivX encoding does not crop the picture to remove
black bands, should they exist. This has quite a negative
impact on bit rate, size and overall picture quality. If you really
want a high-quality, small-size MPEG-4, you should back it up manually.
The improved picture quality is well worth the trouble.
Figure 3. The information bar shows the program name, running TV show and what's
on next.
VDR splits its recordings into 2GB files, which is a bit inconvenient
for transcoding the videos. If you go for manual conversion, which
gives you finer control over the quality/size aspect, mencoder or
transcode are good options. Use the speedy mencoder, which I found to
be perfect for backups to MPEG-4, or transcode, which comes with a lot of
tools. If you favor the I-don't-want-to-care approach, get a hold
of VDRCONVERT.
The README file offers a pretty simple approach to
installing it, and at least you can watch some TV while downloading and
compiling.
With VDRCONVERT you have to
change some scripts and configuration files to adapt the DVD/(S)VCD
resolutions to NTSC, in case PAL is not used where you live.
It's too bad that a Linux PVR doesn't make the TV
programs themselves any better, but I guess you can't have
everything, can you?
Christian A. Herzog is a programmer focused on Web development using
open-source technologies. He's still on his never-ending quest to bring a
Linux-based device to every home and company he comes across.
Write him at [email protected].
Comment viewing options
I was wonder if you could also include a connection diagram . I was looking at the nexus-s card and didn't see a TV out, just a loop connection is this the connection used for the TV. Or are using additional card to get the TV output? |
package x509util
import (
"crypto/rand"
"crypto/rsa"
"crypto/x509"
"crypto/x509/pkix"
"testing"
)
func TestCreateCertificateRequest(t *testing.T) {
r := rand.Reader
priv, err := rsa.GenerateKey(r, 1024)
if err != nil {
t.Fatal(err)
}
template := CertificateRequest{
CertificateRequest: x509.CertificateRequest{
Subject: pkix.Name{
CommonName: "test.acme.co",
Country: []string{"US"},
},
},
ChallengePassword: "foobar",
}
derBytes, err := CreateCertificateRequest(r, &template, priv)
if err != nil {
t.Fatal(err)
}
out, err := x509.ParseCertificateRequest(derBytes)
if err != nil {
t.Fatalf("failed to create certificate request: %s", err)
}
if err := out.CheckSignature(); err != nil {
t.Errorf("failed to check certificate request signature: %s", err)
}
challenge, err := ParseChallengePassword(derBytes)
if err != nil {
t.Fatalf("failed to parse challengePassword attribute: %s", err)
}
if have, want := challenge, template.ChallengePassword; have != want {
t.Errorf("have %s, want %s", have, want)
}
}
|
Dymas
In Greek mythology, Dymas (Ancient Greek: Δύμας) is the name attributed to the following individuals:
Dymas, a Mariandynian who warned the Argonauts about the cruelty of Amycus, king of the Bebrycians. Both Mariandynians and Bebrycians lived in northwestern Asia Minor.
Dymas, a soldier who fought on the side of the Seven Against Thebes. He took part in the foot-race at Opheltes' funeral games in Nemea. Dymas was wounded in battle and killed himself when the enemy started questioning him.
Dymas, a Dorian and the ancestor of the Dymanes. His father, Aegimius, adopted Heracles' son, Hyllas. Dymas and his brother, Pamphylus, submitted to Hyllas.
Dymas, king of Phrygia and father of Hecuba.
Dymas, perhaps the same as the first. According to Quintus Smyrnaeus this Dymas was the father of Meges, a Trojan whose sons fought at Troy.
Dymas, an Aulian warrior, who came to fight at Troy under the leadership of Archesilaus. He died at the hands of Aeneas.
Dymas, a Trojan soldier who fought with Aeneas and was killed at Troy.
Dymas, was mentioned in Homer's Odyssey as a Phaeacian captain, whose daughter was a friend to the princess Nausicaa.
References
Category:Kings of Phrygia
Category:Characters in Greek mythology
Category:Dorian mythology |
MacEwan International
MacEwan International promotes an internationally informed and cross-culturally sensitive learning environment. Our vision is to be a leader in internationalization, preparing all students, as well as faculty and staff, to succeed in and contribute to a global society and economy as members of an interconnected world community. |
Robinsons ready to roll
Twins Tyrell and Tyree Robinson making their marks in football, basketball
A quick look at twin brothers Tyree and Tyrell Robinson (San Diego/Lincoln) reveals one big misconception: They're not identical. Tyree, older by two minutes, is taller by a full inch and likes to distinguish himself by wearing headbands. Tyrell is bulkier and less flashy with his wardrobe choices.
And when the dual-sport athletes take the football field, their differences continue to stack up. |
// Copyright 2000-2020 JetBrains s.r.o. Use of this source code is governed by the Apache 2.0 license that can be found in the LICENSE file.
package com.intellij.openapi.vcs.impl
import com.intellij.ProjectTopics
import com.intellij.openapi.application.ApplicationManager
import com.intellij.openapi.components.service
import com.intellij.openapi.extensions.ExtensionNotApplicableException
import com.intellij.openapi.module.Module
import com.intellij.openapi.module.ModuleManager
import com.intellij.openapi.project.ModuleListener
import com.intellij.openapi.project.Project
import com.intellij.openapi.project.rootManager
import com.intellij.openapi.roots.ModuleRootEvent
import com.intellij.openapi.roots.ModuleRootListener
import com.intellij.openapi.startup.StartupActivity
import com.intellij.openapi.vcs.AbstractVcs
import com.intellij.openapi.vcs.ProjectLevelVcsManager
import com.intellij.openapi.vcs.VcsDirectoryMapping
import com.intellij.openapi.vfs.VirtualFile
internal class ModuleVcsDetector(private val project: Project) {
private val vcsManager by lazy(LazyThreadSafetyMode.NONE) {
(ProjectLevelVcsManager.getInstance(project) as ProjectLevelVcsManagerImpl)
}
internal class MyPostStartUpActivity : StartupActivity.DumbAware {
init {
if (ApplicationManager.getApplication().isUnitTestMode) {
throw ExtensionNotApplicableException.INSTANCE
}
}
override fun runActivity(project: Project) {
val vcsDetector = project.service<ModuleVcsDetector>()
val listener = vcsDetector.MyModulesListener()
val busConnection = project.messageBus.connect()
busConnection.subscribe(ProjectTopics.MODULES, listener)
busConnection.subscribe(ProjectTopics.PROJECT_ROOTS, listener)
if (vcsDetector.vcsManager.needAutodetectMappings()) {
vcsDetector.autoDetectVcsMappings(true)
}
}
}
private inner class MyModulesListener : ModuleRootListener, ModuleListener {
private val myMappingsForRemovedModules: MutableList<VcsDirectoryMapping> = mutableListOf()
override fun beforeRootsChange(event: ModuleRootEvent) {
myMappingsForRemovedModules.clear()
}
override fun rootsChanged(event: ModuleRootEvent) {
myMappingsForRemovedModules.forEach { mapping -> vcsManager.removeDirectoryMapping(mapping) }
// the check calculates to true only before user has done any change to mappings, i.e. in case modules are detected/added automatically
// on start etc (look inside)
if (vcsManager.needAutodetectMappings()) {
autoDetectVcsMappings(false)
}
}
override fun moduleAdded(project: Project, module: Module) {
myMappingsForRemovedModules.removeAll(getMappings(module))
autoDetectModuleVcsMapping(module)
}
override fun beforeModuleRemoved(project: Project, module: Module) {
myMappingsForRemovedModules.addAll(getMappings(module))
}
}
private fun autoDetectVcsMappings(tryMapPieces: Boolean) {
if (vcsManager.haveDefaultMapping() != null) return
val usedVcses = mutableSetOf<AbstractVcs?>()
val detectedRoots = mutableSetOf<Pair<VirtualFile, AbstractVcs>>()
val roots = ModuleManager.getInstance(project).modules.flatMap { it.rootManager.contentRoots.asIterable() }.distinct()
for (root in roots) {
val moduleVcs = vcsManager.findVersioningVcs(root)
if (moduleVcs != null) {
detectedRoots.add(Pair(root, moduleVcs))
}
usedVcses.add(moduleVcs) // put 'null' for unmapped module
}
val commonVcs = usedVcses.singleOrNull()
if (commonVcs != null) {
// Remove existing mappings that will duplicate added <Project> mapping.
val rootPaths = roots.map { it.path }.toSet()
val additionalMappings = vcsManager.directoryMappings.filter { it.directory !in rootPaths }
vcsManager.setAutoDirectoryMappings(additionalMappings + VcsDirectoryMapping.createDefault(commonVcs.name))
}
else if (tryMapPieces) {
val newMappings = detectedRoots.map { (root, vcs) -> VcsDirectoryMapping(root.path, vcs.name) }
vcsManager.setAutoDirectoryMappings(vcsManager.directoryMappings + newMappings)
}
}
private fun autoDetectModuleVcsMapping(module: Module) {
if (vcsManager.haveDefaultMapping() != null) return
val newMappings = mutableListOf<VcsDirectoryMapping>()
for (file in module.rootManager.contentRoots) {
val vcs = vcsManager.findVersioningVcs(file)
if (vcs != null && vcs !== vcsManager.getVcsFor(file)) {
newMappings.add(VcsDirectoryMapping(file.path, vcs.name))
}
}
if (newMappings.isNotEmpty()) {
vcsManager.setAutoDirectoryMappings(vcsManager.directoryMappings + newMappings)
}
}
private fun getMappings(module: Module): List<VcsDirectoryMapping> {
return module.rootManager.contentRoots
.mapNotNull { root -> vcsManager.directoryMappings.firstOrNull { it.directory == root.path } }
}
}
|
Enhanced absorption and inhibited metabolism of emodin by 2, 3, 5, 4'-tetrahydroxystilbene-2-O-β-D-glucopyranoside: Possible mechanisms for Polygoni Multiflori Radix-induced liver injury.
Polygoni Multiflori Radix (PMR) has been commonly used as a tonic in China for centuries. However, PMR-associated hepatotoxicity is becoming a safety issue. In our previous in vivo study, an interaction between stilbenes and anthraquinones has been discovered and a hypothesis is proposed that the interaction between stilbene glucoside-enriching fraction and emodin may contribute to the side effects of PMR. To further support our previous in vivo results in rats, the present in vitro study was designed to evaluate the effects of 2, 3, 5, 4'-tetrahydroxystilbene-2-O-β-D-glucopyranoside (TSG) on the cellular absorption and human liver microsome metabolism of emodin. The obtained results indicated that the absorption of emodin in Caco-2 cells was enhanced and the metabolism of emodin in human liver microsomes was inhibited after TSG treatment. The effects of the transport inhibitors on the cellular emodin accumulation were also examined. Western blot assay suggested that the depressed metabolism of emodin could be attributed to the down-regulation of UDP-glucuronosyltransferases (UGTs) 1A8, 1A10, and 2B7. These findings definitively demonstrated the existence of interaction between TSG and emodin, which provide a basis for a better understanding of the underlying mechanism for PMR-induced liver injury. |
Friends of the Crow Collection: Adults/ Children ($10/ $3) || General Public: Adults/ Children ($18/ $5)
Otsukimi Celebration, 2012
The Japan America Society celebrates the full autumn moon each year with an outdoor picnic, Japanese music, and haiku poetry. Although not commonly observed in modern-day Japan, the moon viewing tradition dates back to the Heian Period (794–A.D. 1185), when the evening was marked with poetry and music by court aristocrats. The celebration later spread to warriors, townspeople, and farmers, and became a harvest festival.
Bring a picnic supper, beverage, and something to sit on as no food or drink will be sold at the event or pre-order an Obento from Mr. Sushi for $18 when purchasing your celebration tickets. Alcohol is not allowed at Winfrey Point, a City of Dallas park facility. For more information, visit jasdfw.org. |
Probing molecules in integrated silicon-molecule-metal junctions by inelastic tunneling spectroscopy.
Molecular electronics has drawn significant attention for nanoelectronic and sensing applications. A hybrid technology where molecular devices are integrated with traditional semiconductor microelectronics is a particularly promising approach for these applications. Key challenges in this area include developing devices in which the molecular integrity is preserved, developing in situ characterization techniques to probe the molecules within the completed devices, and determining the physical processes that influence carrier transport. In this study, we present the first experimental report of inelastic electron tunneling spectroscopy of integrated metal-molecule-silicon devices with molecules assembled directly to silicon contacts. The results provide direct experimental confirmation that the chemical integrity of the monolayer is preserved and that the molecules play a direct role in electronic conduction through the devices. Spectra obtained under varying measurement conditions show differences related to the silicon electrode, which can provide valuable information about the physics influencing carrier transport in these molecule/Si hybrid devices. |
Promethium: uses
The following uses for promethium are gathered from a number of sources as well as from anecdotal comments. I'd be delighted to receive corrections as well as additional referenced uses (please use the feedback mechanism to add uses).
shows promise as a portable X-ray unit
possibly useful as a heat source to provide auxilliary power for space probes and satellites |
Prevalence of varicocoele and its association with body mass index among 39,559 rural men in eastern China: a population-based cross-sectional study.
Varicocoele is a common cause of male infertility. We undertook a population-based cross-sectional study to evaluate the prevalence of varicocoele among rural men in eastern China and its association with body mass index. A total of 39,559 rural men in six counties in Beijing, Guangdong and Shandong provinces were recruited from 2011 to 2012. The presence and severity of varicocoele were measured by physical examinations. Univariate and multivariate logistic regression models were constructed to assess the association between varicocoele and body mass index after adjusting for possible confounders. Varicocoele was diagnosed in 1911 of 39,559 participants with an overall prevalence of 4.83%. The prevalence of varicocoele was highest in underweight (6.29%) and lowest in obese patients (3.71%, p < 0.05). The prevalence also decreased as body mass index increased in all three varicocoele grades. In multivariate logistic regression analysis after adjusting for region, age, height, occupation, cigarette smoking and alcohol consumption, body mass index was still inversely and independently associated with varicocoele (p < 0.001). Compared with normal weight men, underweight men (OR = 1.34; 95% CI, 1.10-1.63) were more likely to have varicocoele, whereas overweight men (OR = 0.88; 95% CI, 0.79-0.99) and obese men (OR = 0.75; 95% CI, 0.58-0.97) were less likely to have varicocoele. This study revealed that the prevalence of varicocoele was 4.83% among rural men in eastern China; body mass index was inversely and independently associated with the presence of varicocoele. Future efforts should be made to validate the risk factors for varicocoele and strengthen the prevention and treatment of varicocoele, especially in underweight men. |
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The Huffington Post: Top Design Destinations for 2017
2017-02-23
By Janette Ewen
Ever since Frank Gehry’s spectacular Guggenheim Bilbao put its sleepy namesake city on the radar of architecture buffs two decades ago, design has became an integral aspect of travel and tourism, joining food, culture and climate when it comes to visitor draws. This year, the list of destinations sure to entice design fans includes spots from the West Indies to North Africa. They offer a wide range of aesthetic attractions, from cutting-edge urban design to exquisite historical gems.
OLD HAVANA, NEW URGENCY
Whether the recent detente between the United States and Cuba will result in an onslaught of American visitors to the island or not, Canadians aren’t waiting to find out: According to KAYAK, a world-leading travel search engine, Havana is one of the year’s top 10 trending destinations among travellers from the Great White North, whose online inquiries about the city skyrocketed by 230 percent compared to last year. In anticipation of more visitors, hotels in Havana are being modernized and restaurants given new polish, but it’s the bustling metropolis’ status as a living design museum that no doubt appeals to most foreigners. For architecture fans, hotels like the Nacional offer glimpses into long-gone eras, while automobile buffs would be hard-pressed to find a greater parade of vintage cars. Speaking of moveable feasts, bars like La Floridita, where Ernest Hemingway indulged his fondness for daiquiris, are modern-day links to literary and artistic legends. Clearly, the time to visit Havana is now, whatever your aesthetic bent.
CARIBBEAN COOL
Over the past several years, restaurant-rich Grand Cayman, the largest of the Cayman Islands, has been nurturing a reputation as the culinary capital of the Caribbean. Now, its growing foodie cred is being matched by its design cachet. In November, the ultra-sleek Kimpton Seafire Resort + Spa, designed by U.S. firm SB Architects, opened on Seven Mile Beach, bringing a welcome shot of global chic (plus four more dining options) to that pristine stretch of coastline. Not far away, Camana Bay, an ambitious mixed-use development, has been heralded as a rare example of new urbanism in the region, its 500 acres encompassing high-end shops, office and residential space, interactive fountains and a pedestrianized main street called the Paseo. Situated between the Kimpton and Camana Bay is the Caribbean Club, a luxury apartment hotel and ideal base for exploring the area; it also houses one of Grand Cayman’s foremost eateries, the trattoria Luca.
ROAD TO MOROCCO
Another top trender among Canadian travellers according to KAYAK is Casablanca, the romantic Moroccan city that has long offered a beguiling mix of French and Arabic cultures. Nowhere is this hybrid allure more visible than in its architecture, which ranges from the art deco elegance of Place Mohammed V to contemporary showstoppers like the Four Seasons Casablanca on the oceanfront Corniche. At bustling Marche Centrale, the Moorish-style setting is as enticing as the fried fish and grilled vegetables, while L’Atelier 21, the city’s leading modern art gallery, showcases emerging and established artists in an au courant space. New air links to Casablanca from Canada this year make visiting even easier.
LONDON CALLING
The British capital has always been a magnet for design aficionados, but 2017 offers an extra-special reason to visit: the recently transplanted Design Museum, which has been moved from its previous home on the south bank of the Thames to much larger digs in Kensington. Ten years in the making, the $140-million wood-and-concrete marvel, reimagined by minimalist architect John Pawson on the site of the former Commonwealth Institute, is the Brit superstar’s first public building in London. Visitors must pay to see special exhibitions, but the museum’s extensive permanent collection, which includes everything from a 2012 Olympic torch to a full-size Tube car, is free to view. Another area museum completing a major update this year is the venerable Victoria and Albert, which will unveil a new underground gallery and a new entrance on Exhibition Road in July. Even the city’s best watering holes are offering new eye candy: Check out the restored blue walls in The Berkeley’s expanded Blue Bar. |
Mai-Mai
The term Mayi-Mayi or Mai-Mai refers to any kind of community-based militia group active in the Democratic Republic of the Congo (DRC), formed to defend their local territory against other armed groups. Most were formed to resist the invasion of Rwandan forces and Rwanda-affiliated Congolese rebel groups, but some may have formed to exploit the war for their own advantage by looting, cattle rustling or banditry.
Groups that fall under the umbrella term "Mai-Mai" include armed forces led by warlords, traditional tribal elders, village heads, and politically motivated resistance fighters. Because Mai Mai have had only the most tenuous internal cohesion, different Mai-Mai groups allied themselves with a variety of domestic and foreign government and guerrilla groups at different times. The term Mai-Mai does not refer to any particular movement, affiliation or political objective but to a broad variety of groups.
Mai-Mai were particularly active in the eastern Congolese provinces bordering Rwanda, North Kivu and South Kivu (the "Kivus"), which were under the control of the Rwanda-allied Banyamulenge-dominated rebel faction, the Rally for Congolese Democracy–Goma (RCD-Goma) during the Second Congo War. While militias have long been common in the Kivus, particularly among the minority Batembo and Babembe ethnic groups, the recent wars and conflicts caused large numbers of town dwellers to form Mai-Mai. Although the Mai-Mai, either as a group or as individual groups, were not party to the 1999 Lusaka Accord meant to end the Second Congo War, they remained one of the most powerful forces in the conflict and the lack of cooperation from some groups has been problematic for the peace process.
Mai-Mai in North and South Kivu
According to a 2001 UN report, 20,000 to 30,000 Mai-Mai were active in the two Kivu provinces. The two most powerful and well-organized Mai-Mai groups in the Kivus were led by Generals Padiri and Dunia. Currently most active is a group which is called Mai-Mai Yakutumba, was organized in 2007 by General Yakutumba. They were reported to have received aid from the government of the Democratic Republic of Congo and are widely viewed by other Mai Mai groups as the leaders, though not the commanders, of the Kivu Mai-Mai. A number of smaller Mai-Mai groups, such as the Mudundu 40/Front de Résistance et de Défense du Kivu (FRDKI) and Mouvement de Lutte contre l'Agression au Zaïre/Forces Unies de Résistance Nationale contre l'Agression de la Républíque Démocratique du Congo (MLAZ/FURNAC), were reported to cooperate with the Rwandan military and Rally for Congolese Democracy–Goma (RCD-Goma).
Walikale and Masisi north of Goma were the centres of Mai-Mai activity in North Kivu. In South Kivu, there have historically been concentrations around Walungu and Bunyakiri south of Lake Kivu, around Uvira and Mwenaga at the northern end of Lake Tanganyika, further south around Fizi, and around Shabunda, between the Rwandan border and Kindu.
A Mai-Mai leader, Colonel Mayele, was arrested by UN forces in October 2010, allegedly being the leader behind mass rapes in the Walikale region of North Kivu province.
Mai-Mai in Katanga
A former leader of the Mai-Mai, Gédéon Kyungu Mutanga, turned himself over to MONUC troops in May 2006. He was found guilty of numerous war crimes between October 2003 and May 2006 and was sentenced to death by the Kipushi Military Tribunal in Katanga Province on 6 March 2009. He escaped from prison in September 2011 and formed the Mai-Mai Kata Katanga ("Secede Katanga").
Other Mai-Mai groups
There was a large Mai-Mai presence in Maniema, in particular around Kindu and Kalemie. Province Orientale also hosts a number of Mai-Mai, but these groups were apparently involved in long-standing ethnic disputes.
Mai-Mai Gedeon is also commanded by Gedeon Kyungu Mutanga and loosely tied to his Mai-Mai Kata Katanga. The Corak Kata Katanga also known as the Co-ordination for a Referendum on Self-determination for Katanga, composed mainly of former Katanga Tigers, a separatist group active in the 1960s. They claim to be behind the attack on the Katanga airport in February 2011. It is unclear to what extent all these groups are co-ordinated.
The Nduma Defense of Congo (or Mai-Mai Sheka) was formed in 2009 by former minerals trader Ntabo Ntaberi Sheka, an ethnic Nyanga. Sheka claims the group was formed to liberate the mines of Walikale Territory in North Kivu. The NDC are accused of a mass rape of at least 387 women, men, and children over a three day span in Walikale in 2010.
Mai-Mai and the mountain gorillas
In May 2007, Mai-Mai killed two wildlife officers in Virunga National Park and threatened to kill mountain gorillas if the government retaliated. The Mai-Mai are also suspected of the killings of nine mountain gorillas, with the use of machetes, and automatic weapons. In an October 2012 incident, Mai-Mai killed two park staff and a soldier, while three soldiers were injured. From 1990 to 2018 some 170 Virunga Rangers have died in such attacks, according to the World Wildlife Foundation.
Six Virunga Park Rangers were reported to have been killed in Virunga National Park. Five rangers and a driver were killed in an ambush and a sixth ranger was injured in the Central section of the vast reserve on April 9, 2018. Officials suspected the attacks were by the Mai-Mai.
See also
Resistance Patriots Maï-Maï
Mai-Mai Kata Katanga
Gédéon Kyungu Mutanga
References
External links
Global Security description
UN Assessment of armed groups in Congo, 1 April 2002
National Geographic
Mai-mai atrocities included canibalism
Category:Factions of the Second Congo War
Category:History of the Democratic Republic of the Congo
Category:History of Rwanda
Category:Rebel groups in the Democratic Republic of the Congo
Category:Rebel groups that actively control territory
Category:Vigilantism |
Methylglyoxal activates NF-κB nuclear translocation and induces COX-2 expression via a p38-dependent pathway in synovial cells.
There is growing evidence of an increased prevalence of osteoarthritis (OA) among people with diabetes. Synovial inflammation and increased expression of cyclooxygenase-2 (COX-2) are two key features of patients with OA. Methylglyoxal (MGO) is a common intermediate in the formation of advanced glycation end-products, and its concentration is also typically higher in diabetes. In this study, we investigated the effects of the treatment of different MGO concentrations to rabbit HIG-82 synovial cells on COX-2 expression. The MGO induced COX-2 mRNA expression was detected by quantitative polymerase chain reaction. The MGO induced COX-2 protein production and its signaling pathways were detected by western blotting. The nuclear factor-kappa B (NF-κB) nuclear translocation by MGO was examined by immunofluorescence. In the present study, we find that MGO has no toxic effects on rabbit synovial cells under the experimental conditions. Our analysis demonstrates that MGO induced COX-2 mRNA and protein production. Moreover, MGO induces p38-dependent COX-2 protein expression as well as the phosphorylations of extracellular signal-regulated kinase, c-Jun N-terminal kinase (JNK), and Akt/mammalian target of rapamycin (mTOR)/p70S6K; however, inhibition of JNK and Akt/mTOR/p70S6K phosphorylations further activates COX-2 protein expression. Furthermore, MGO is shown to activate of nuclear factor-kappa B (NF-κB) nuclear translocation. Our results suggest that MGO can induce COX-2 expression via a p38-dependent pathway and activate NF-κB nuclear translocation in synovial cells. These results provide insight into the pathogenesis of the synovial inflammation under the diabetic condition associated with higher MGO levels. |
Q:
Do we want an xkcd tag?
xkcd is referred to often on PPCG, with at least 47 questions which are based on concepts or directly related to the xkcd webcomic.
Therefore, is it worthwhile introducing an xkcd tag to group all of these challenges together?
A:
No
Tags are meant to classify questions according to some distinctive quality that they share. Simply referencing an xkcd comic is not a distinctive quality that would create a meaningful classification.
|
[A meningioma in the posterior fossa without dural attachment: case report].
An extremely rare case of a meningioma in the posterior fossa without dural attachment has been reported. The patient was a 56-year-old male whose chief manifestation was the abnormality of his CT scan. His past history included gastric and colonic polyp when he was 54, 55 years old, and non-Hodgkin's lymphoma before hospitalization in our department. CT scan showed a small round non-enhancing lesion located at the lateral site of the right cerebellar cortex. T1 weighted image of MRI showed a homogeneous low intensity lesion with partial enhancing with Gd-DTPA. Proton image showed a remarkable low intensity lesion which showed an extramedullary mass. Right retromastoid craniectomy was performed. The mass was an extramedullary tumor which had no relation with the cerebellar cortex and dura matter. The arachnoid membrane around the tumor was intact. The tumor was totally resected and the patient had no neurological deficits. Histopathologically, the tumor was delineated into laminar structures by collagen fiber. Tumor cells were spindle in shape and made a whorling formation. There was no psammoma body and it had a hyperchromatic nuclei without mitotic features. Electron microscopic studies revealed no typical interdigitation but irregularity of the cell membrane. Abundant collagen fibers were in contact with basement membrane of the tumor. According to these findings, we diagnosed fibroblastic meningioma with atypical forms. Meningiomas without dural attachment are rare in adults, especially extremely rare of the posterior fossa. There are only 23 previous reports of "meningioma of the posterior fossa without dural attachment". Cantore divided these meningiomas into three groups (IV ventricle, inferior tela choroidea and cisterna magna).(ABSTRACT TRUNCATED AT 250 WORDS) |
Genetic diversity studies of Brazilian garlic cultivars and quality control of garlic-clover production.
The garlic cultivars grown in Brazil evolved from somatic mutations and clone selection by breeding programs and by the introduction of germplasm from other countries. Morphological characters have been used to differentiate these cultivars. Two hundred and six random amplified polymorphic DNA markers were utilized for a diversity analysis of the 17 most planted garlic cultivars in Brazil. Bootstrap analysis showed that the number of markers was efficient and sufficient to obtain a coefficient of variation of 10%. Similarity varied between 16 and 98% and cluster analysis showed that, in general, genetic similarities correlate with morphological characters of the cultivars and production cycle variation. High bootstrap values at most of the nodes supported the dendrogram stability. The grouping of most varieties agreed well with previous reports based on morphological characters. As a vegetative-propagated species, viral diseases are a key problem regarding production and quality of the bulbs, causing gradual loss of yield and decrease in storage capacity. To improve the health quality of garlic seed, a virus-free stock of garlic cloves of the Amarante cultivar was obtained. The ability to distinguish garlic cultivars to detect varietal mixing after in vitro multiplication is extremely important, since correct identification is not possible until bulbs are produced. Random amplified polymorphic DNA markers were also used to differentiate cultivars while they are in vitro and not amenable to morphological discrimination. No difference was identified between the fingerprints of the virus-free or of the infected bulks of Amarante, showing that there was no clove mixing in the handling of material in the clonal multiplication phase. |
Conventional methods for producing metal powder include a water atomizing method, which provides metal powder by injecting a high pressure water jet to a flow of a molten material; a gas atomizing method, which employs spraying of N2 gas or Ar gas in place of the water jet used in the atomizing method; and a centrifugation method, in which a molten material jet is injected into cooling water present in a rotary drum rotating at high speed. Fine particles are also produced through a breakdown method such as mechanical formation employing a mill or the like and also through a buildup method such as a precipitation method or a sol-gel method.
However, in the water atomizing method and the gas atomizing method, the nozzle structure is complicated and an excessive load is imposed on nozzles, resulting in lowered durability of the nozzle, since the molten material is formed into powder form by a flow of high pressure cooling water or cooling gas. Meanwhile, in the centrifugation method, the structure of the apparatus is complicated, in order to enable high-speed rotation of the rotary drum. Furthermore, in these methods, the molten metal is pulverized on the basis of collision energy. Thus, the resulting particle size is varied, and the yield of fine particles is poor. The breakdown method employing mechanical formation or the like can produce only large particles having a minimum size of, for example, approximately 100 μm. The buildup method such as a precipitation method can produce fine particles having a maximum size of approximately 1 μm, and particles which are larger than approximately 1 μm cannot be obtained. Therefore, when conventional methods and apparatuses for producing fine particles are employed, fine particles having a size ranging from several micrometers to the order of 10 μm, particularly fine particles having a size of about 3 μm, are difficult to obtain. Also, in the breakdown method, a large portion of the molten metal cannot be converted into fine particles and remains as a lump, thereby deteriorating the yield thereof. In addition, the particle size distribution assumes a broaden profile, causing the problem that fine particles having a desired particle diameter cannot be obtained in a large amount.
Conventionally, a liquid quenching method has been known for producing amorphous metal. According to the liquid quenching method, a molten material is cooled and solidified by, for example, causing a molten metal liquid to spout into a coolant, whereby amorphous metal is produced. Even when a centrifugation method, which can attain a relatively large cooling rate, is employed in combination with the liquid quenching method, the heat flux between two liquids (i.e., molten material and coolant) is limited to the critical heat flux in the case where heat conduction is induced by cooling based on convection or a conventional boiling method. Thus, the cooling rate is limited to 104 to 105 K/s, which problematically imposes limitation on the type of metal which can be converted into an amorphous material.
Previously, the present applicant filed a patent application for a method for producing fine particles and amorphous material of molten material which includes supplying into a liquid coolant a molten material which has been formed by melting a raw material to be converted into fine particles or amorphous material, with a small difference in flow speed of the two liquids, to thereby cause boiling by spontaneous bubble nucleation and employing the resultant pressure wave for producing fine particles and amorphous material thereof (see Patent Documents: WO 01/81033 and WO 01/81032).
However, according to the method for which the present applicant previously filed a patent application, when a high-melting material having a melting point of, for example, 800° C. or higher is used, vapor film cannot be broken satisfactorily through condensation. Thus, formation of fine particles or amorphous material of molten material cannot be fully achieved.
Thus, an object of the present invention is to provide, on the basis of improvement of the previously developed technique, a method for producing fine particles, the method being capable of producing fine particles from a high-melting-point raw material and readily producing submicron fine particles which have not been readily produced through the previously developed technique. Another object of the invention is to provide an apparatus therefor. |
package io.gitlab.arturbosch.detekt.generator.collection
import io.gitlab.arturbosch.detekt.api.DetektVisitor
import io.gitlab.arturbosch.detekt.generator.collection.exception.InvalidDocumentationException
import io.gitlab.arturbosch.detekt.rules.isOverride
import org.jetbrains.kotlin.psi.KtCallExpression
import org.jetbrains.kotlin.psi.KtClassOrObject
import org.jetbrains.kotlin.psi.KtFile
import org.jetbrains.kotlin.psi.KtProperty
import org.jetbrains.kotlin.psi.KtReferenceExpression
import org.jetbrains.kotlin.psi.KtSuperTypeList
import org.jetbrains.kotlin.psi.KtValueArgumentList
import org.jetbrains.kotlin.psi.psiUtil.containingClass
import org.jetbrains.kotlin.psi.psiUtil.referenceExpression
data class MultiRule(
val name: String,
val rules: List<String> = listOf()
) {
operator fun contains(ruleName: String) = ruleName in this.rules
}
private val multiRule = io.gitlab.arturbosch.detekt.api.MultiRule::class.simpleName ?: ""
class MultiRuleCollector : Collector<MultiRule> {
override val items = mutableListOf<MultiRule>()
override fun visit(file: KtFile) {
val visitor = MultiRuleVisitor()
file.accept(visitor)
if (visitor.containsMultiRule) {
items.add(visitor.getMultiRule())
}
}
}
class MultiRuleVisitor : DetektVisitor() {
val containsMultiRule
get() = classesMap.any { it.value }
private var classesMap = mutableMapOf<String, Boolean>()
private var name = ""
private val rulesVisitor = RuleListVisitor()
private val properties: MutableMap<String, String> = mutableMapOf()
fun getMultiRule(): MultiRule {
val rules = mutableListOf<String>()
val ruleProperties = rulesVisitor.ruleProperties
.mapNotNull { properties[it] }
rules.addAll(ruleProperties)
rules.addAll(rulesVisitor.ruleNames)
if (name.isEmpty()) {
throw InvalidDocumentationException("MultiRule without name found.")
}
if (rules.isEmpty()) {
throw InvalidDocumentationException("MultiRule $name contains no rules.")
}
return MultiRule(name, rules)
}
override fun visitSuperTypeList(list: KtSuperTypeList) {
val isMultiRule = list.entries
?.mapNotNull { it.typeAsUserType?.referencedName }
?.any { it == multiRule } ?: false
val containingClass = list.containingClass()
val className = containingClass?.name
if (containingClass != null && className != null && !classesMap.containsKey(className)) {
classesMap[className] = isMultiRule
}
super.visitSuperTypeList(list)
}
override fun visitClassOrObject(classOrObject: KtClassOrObject) {
super.visitClassOrObject(classOrObject)
if (classesMap[classOrObject.name] != true) {
return
}
name = classOrObject.name?.trim() ?: ""
}
override fun visitProperty(property: KtProperty) {
super.visitProperty(property)
if (classesMap[property.containingClass()?.name] != true) {
return
}
if (property.isOverride() && property.name != null && property.name == "rules") {
property.accept(rulesVisitor)
} else {
val name = property.name
val initializer = property.initializer?.referenceExpression()?.text
if (name != null && initializer != null) {
properties[name] = initializer
}
}
}
}
class RuleListVisitor : DetektVisitor() {
var ruleNames: MutableSet<String> = mutableSetOf()
private set
var ruleProperties: MutableSet<String> = mutableSetOf()
private set
override fun visitValueArgumentList(list: KtValueArgumentList) {
super.visitValueArgumentList(list)
val argumentExpressions = list.arguments.map { it.getArgumentExpression() }
// Call Expression = Constructor of rule
ruleNames.addAll(argumentExpressions
.filterIsInstance<KtCallExpression>()
.map { it.calleeExpression?.text ?: "" })
// Reference Expression = variable we need to search for
ruleProperties.addAll(argumentExpressions
.filterIsInstance<KtReferenceExpression>()
.map { it.text ?: "" })
}
}
|
Failure of covalently cross-linked human IgG myeloma subclass protein to release histamine from human leukocytes.
We have examined the ability of IgG subclass antibodies to release histamine from human leukocytes using covalently cross-linked oligomers of human myeloma proteins. Purified IgG1, G2, G3, G4, (or IgE) was incubated with dimethyl suberimidate to induce cross-linking. The resulting dimers, trimers, and higher molecular weight oligomers were isolated using gel filtration columns (Sephadex G200 and Ultrogel AcA 22) connected in tandem. None of the oligomers of IgG1, G2, G3, or G4 released histamine from leukocytes of donors whose basophils released histamine when challenged with IgE dimer. Furthermore, preincubation with subclass specific oligomers did not desensitize cells to challenge with IgE dimer or to anti-IgE. We conclude that, under our experimental conditions, oligomers of human IgG myeloma subclass antibodies do not trigger histamine release nor modulate IgE-mediated reactions. |
Those words (choice profanity included) woke me with a start the other night. What was I thinking, organizing this trip to Vietnam to connect sons and daughters who lost fathers on both sides of the Vietnam War?
I have a lot of fears about this journey. There’s the mundane ones about getting sick, or bitten by something slimy. Maybe I'll become separated from the group because something in a shop caught my eye (this, given my nature, is the most likely scenario). But the deeper fears are right under the surface. What’s going to happen when we come face to face with the Vietnamese sons and daughters? Will they be angry? Worse, will I?
It was easy to push past these bigger fears earlier this year when I first formed the 2 Sides Project. Now the trip is getting closer—we leave four weeks from today—and they’re keeping me up at night.
I’m going to have to remember what I know in the daylight: there have been moments in my life when I’ve found people who shared my experience, who spoke the same language as me, who felt the same way I did about things. These moments are profound. They make me feel connected, anchored in the world. They are often turning points that lead me to a better place.
That was the case when I met other sons and daughters in the U.S. who lost fathers in the war. So, I’ll keep my focus on them. And on the amazing experience we have in store. Six of us will be meeting Vietnamese sons and daughters and visiting the sites where our fathers died. I’ll profile them all -- Mike, Ron, Margaret, Susan and Patty -- here in the coming weeks as we get ready. Come with us virtually. It’s going to be quite a journey, and we’re looking forward to sharing it with you. |
Cognitive and behavioral effects of carbamazepine in children: data from benign rolandic epilepsy.
The effects of antiepileptic drugs on cognition are difficult to delineate, yet of critical importance for children with epilepsy. We investigated the cognitive and behavioral effects of carbamazepine in children with benign rolandic epilepsy. Ten subjects with benign rolandic epilepsy were evaluated with and without carbamazepine treatment. Fourteen unmedicated subjects with migraine headache evaluated twice served as a control group. Subjects were 6 to 12 years of age, fluent in English, and not mentally retarded. We found that children with benign rolandic epilepsy were quicker on a visual-search task and recalled stories better when not treated than when treated with carbamazepine. After correction for multiple comparisons only the memory finding remained significant. Higher carbamazepine serum level was associated with slower performance on the same visual-search task. This latter finding did not meet multiple comparison criteria. Numerous significant practice effects were found within the control group. Comparisons with reliable change indices identified two subjects with benign rolandic epilepsy with particularly poor scores while receiving carbamazepine. These findings suggest some effects on memory from carbamazepine; however, they do not support meaningful dosage-related effects, within the recommended range. Significant practice effects confirmed the need to control for such effects when evaluating treatments. Finally, identification of two subjects who performed more poorly while on carbamazepine suggests that some children might experience particular difficulties while receiving this medication and highlights the need to investigate individual subject responses to treatment. |
Got this cute little sewing chair from Sara and Stacy at SugarSCOUT–they have “super sweet finds of all kinds”…just check out their Etsy Shop. (lots of great ideas on their blog @ www.sugarSCOUT.com, too!)
I do love spending time in my studio that has become a haven for creating my upcycled bags.
I’m adding new bags as quick as I get them done to my Etsy shop. Take a look…it’s called itzaChicThing.
I love to layer color, pattern and texture. I created this bag using a fusing process. After making many bags, all shapes and sizes (you can see some of them at bohochicbag.com), I decided to use the same concept to create pieces for hanging. |
This invention relates generally to enzymes that convert sucrose to isomaltulose. More particularly, the present invention relates to novel sucrose isomerases, to polynucleotides encoding these enzymes, to methods for isolating such polynucleotides and to nucleic acid constructs that express these polynucleotides. The invention also relates to cells, particularly transformed bacterial or plant cells, and to differentiated plants comprising cells, which express these polynucleotides. The invention further relates to the use of the polypeptides, polynucleotides, cells and plants of the invention for producing isomaltulose.
Isomaltulose α-D-glucopyranosyl-1,6-D-fructofuranose (also called palatinose) is a naturally occurring structural isomer of sucrose (α-D-glucosyl-1,2-D-fructose). Isomaltulose is a nutritive disaccharide, with sweetness and bulk similar to sucrose. Several characteristics make isomaltulose advantageous over sucrose for some applications in the food industry: 1) noncariogenic (not causing dental decay); 2) low glycemic index (useful for diabetics); 3) selective promotion of growth of beneficial bifidobacteria among human intestinal microflora; 4) greater stability of isomaltulose-containing foods and beverages; 5) less hygroscopic; 6) simple conversion into sugar alcohols with other useful properties as foods. The safety of isomaltulose has been comprehensively verified, resulting in unqualified approval as human food, and it is widely used commercially as a sucrose substitute in foods, soft drinks and medicines (Takazoe, 1989, Palatinose—an isomeric alternative to sucrose. In: Progress in Sweeteners (T H Grengy, ed.) pp 143-167. Elsevier, Barking, UK).
Furthermore, because isomaltulose has an accessible carbonyl group, it has attracted attention as a renewable starting material for the manufacture of bioproducts such as polymers and surfactants with potential advantages over substances manufactured from petroleum (Cartarius et al., 2001, Chemical Engineering and Technology 24: 55A-59A; Kunz, 1993, From sucrose to semisynthetical polymers. In: Carbohydrates as Organic Raw Materials II (G Descotes, ed.) pp 135-161. VCH, Weinheim; Lichtenthaler et al., 2001, Green Chemistry 3: 201-209; Schiweck et al., 1991, New developments in the use of sucrose as an industrial bulk chemical. In: Carbohydrates as Organic Raw Materials (F W Lichtenthaler, ed.) pp 57-94. VCH, Weinheim).
Commercial isomaltulose is produced from food-grade sucrose by enzymatic rearrangement from a (1,2)-fructoside to a (1,6)-fructoside followed by crystallization. Sucrose isomerase (SI) enzymes (also known as isomaltulose synthases), which are able to convert sucrose to isomaltulose, have been demonstrated in Protaminobacter rubrum, Erwinia rhapontici, E. carotovora var atroseptica, Serratia plymuthica, S. marcesens, Pseudomonas mesoacidophila, Leuconostoc mesenteroides, Klebsiella spp., Agrobacterium sp., haploid yeast and Enterobacter sp. (Avigad 1959, Biochemical Journal 73: 587-593; Bornke et al., 2001, Journal of Bacteriology 183: 2425-2430; Cheetham et al., 1982 Nature 299: 628-631; Huang et al., 1998, Journal of Industrial Microbiology & Biotechnology 21: 22-27; Lund and Waytt, 1973, Journal of General Microbiology 78: 331-3; Mattes et al., 1998, U.S. Pat. No. 5,786,140; McAllister et al., 1990, Biotechnology Letters 12: 667-672; Miyata et al., 1992, Bioscience Biotechnology and Biochemistry 56: 1680-1681; Munir et al., 1987, Carbohydrate Research 164: 477-485; Nagai et al., 1994, Bioscience Biotechnology and Biochemistry 58: 1789-1793; Nagai-Miyata et al., 1993, Bioscience Biotechnology and Biochemistry 57: 2049-2053; Park et al., 1996, Revista De Microbiology 27: 131-136; Schmidt-Berg-Lorenz and Maunch, 1964, Zeitung fur die Zuckerindustrie 14: 625-627; Stotola et al., 1956, Journal of the American Chemical Society 78: 2514-2518; Tsuyuki et al., 1992, Journal of General and Applied Microbiology 38: 483-490; Zhang et al., 2002, Applied and Environmental Microbiology 68: 2676-2682). Isomaltulose is currently produced in industrial scale column reactors containing immobilized bacterial cells. Initially, natural isolates have been used for this purpose but it is anticipated that higher yields of isomaltulose may be achieved using recombinant techniques. Mattes et al. (1998, supra) disclose isolated polynucleotides from Protaminobacter rubrum (CBS 547,77), Erwinia rhapontici (NCPPB 1578), the microorganism SZ 62 (Enterobacter species) and the microorganism MX-45 (Pseudomonas mesoacidophila FERM 11808 or FERM BP 3619) for producing recombinant partial or full-length sucrose isomerase enzymes in host cells such as Escherichia coli. Mattes et al. also disclose conserved amino acid sequences for designing degenerate oligonucleotides for cloning sucrose isomerase-encoding polynucleotides by the polymerase chain reaction (PCR).
In addition to isomaltulose, reported SIs produce varying proportions of the isomer trehalulose (1-O-α-D-glucopyranosyl-D-fructose) along with glucose and fructose as by-products. Some purified SIs produce predominantly isomaltulose (75-85%), others predominantly trehalulose (90%). The ratio of these products varies with reaction conditions, particularly temperature and pH, and under some conditions small quantities of other products such as isomaltose and isomelezitose may be formed (Véronèse and Perlot, 1999, Enzyme and Microbial Technology 24: 263-269). The formation of multiple products lowers the yield and complicates the recovery of the desired isomer. Slow conversion of sucrose into isomaltulose, and a narrow range of optimal reaction conditions also limit the industrial efficiency of isomaltulose production (Cheetham, 1984, Biochemical Journal 220: 213-220; Schiweck et al., 1990, Zuckerindustrie 115: 555-565.). An ideal SI would show high speed, complete conversion, high specificity and a wide window of reaction conditions for isomaltulose production. |
Twist relates to tubular epithelial-mesenchymal transition and interstitial fibrogenesis in the obstructed kidney.
Epithelial-mesenchymal transition (EMT) is a critical step in renal fibrosis. It has been recently reported that a transcription factor, Twist, plays a pivotal role in metastasis of breast tumors by inducing EMT. In this study, we examined whether Twist relates to renal fibrogenesis including EMT of tubular epithelia, evaluating Twist expression level in the unilateral ureteral obstruction (UUO) model. Kidneys of mice subjected to UUO were harvested 1, 3, 7, and 10 days after obstruction. Compared with control kidneys, Twist mRNA-level significantly increased 3 days after UUO (UUO day 3 kidney) and further augmented until 10 days after UUO. Twist expression increased in tubular epithelia of the dilated tubules and the expanded interstitial areas of UUO kidneys, where cell-proliferating appearances were frequently found in a time-dependent manner. Although a part of tubular cells in whole nephron segment were immunopositive for Twist in UUO day 7 kidneys, tubular epithelia downstream of nephron more frequently expressed Twist than upstream of nephron. In UUO day 7 kidneys, some tubular epithelia were confirmed to coexpress Twist and fibroblast-specific protein-1, a marker for EMT, indicating that Twist is involved in tubular EMT under pathological state. Twist was expressed also in a number of alpha-smooth muscle actin-positive myofibroblasts located in the expanded interstitial area of UUO kidneys. From these findings, the present investigation suggests that Twist is associated with tubular EMT, proliferation of myofibroblasts, and subsequent renal fibrosis in obstructed kidneys. |
[Justin Timberlake & Chris Stapleton:]
Sometimes the greatest way to say something is to say nothing at all
Sometimes the greatest way to say something is to say nothing at all
Sometimes the greatest way to say something is to say nothing
But I can't help myself, no I can't help myself, no, no
Caught up in the middle of it
No I can't help myself, no I can't help myself, no, no, no
Caught up in the rhythm of it
[Justin Timberlake & Chris Stapleton:]
Sometimes the greatest way to say something is to say nothing at all
Sometimes the greatest way to say something is to say nothing at all
Sometimes the greatest way to say something is to say nothing |
Q:
Binding value to select in angular js across 2 controllers
Working with angularJS I am trying to figure out a way to bind the value of a select element under the scope of controller A to use it as an argument for an ng-click call [getQuizByCampID() Function] under the scope of controller B.
My first idea was to use jquery, but I have read in the link below that using jquery is not recommended when starting with angularJS.
"Thinking in AngularJS" if I have a jQuery background?
I also read in the link below that this is performed using ng-model, the only problem is that that the example provided is all under the same controller.
and Binding value to input in Angular JS
What is the angularJS way to get the value of the select element under controller A into the function call in the select under controller B?
Price.html view
<div class="col-sm-3" ng-controller="campCtrl"> **Controller A**
<select id="selCampID" class="form-control" ng-model="campInput" >
<option ng-repeat="camp in campaigns" value="{{camp.camp_id}}">{{camp.camp_name}}</option>
</select>
</div>
<div class="col-sm-3" ng-controller="quizCtrl"> **Controller B**
<select ng-click="getQuizByCampID($('#selCampID').val())" class="form-control" ng-model="quizInput">
<option ng-controller="quizCtrl" ng-repeat="quiz in quizzesById" value="{{quiz.quiz_id}}">{{quiz.quiz_name}}</option>
</select>
</div>
App.js
var app= angular.module('myApp', ['ngRoute']);
app.config(['$routeProvider', function($routeProvider) {
$routeProvider.when('/price', {templateUrl: 'partials/price.html', controller: 'priceCtrl'});
}]);
$routeProvider.when('/price', {templateUrl: 'partials/price.html', controller: 'priceCtrl'});
Quiz Controller
'use strict';
app.controller('quizCtrl', ['$scope','$http','loginService', function($scope,$http,loginService){
$scope.txt='Quiz';
$scope.logout=function(){
loginService.logout();
}
getQuiz(); // Load all available campaigns
function getQuiz(campID){
$http.post("js/ajax/getQuiz.php").success(function(data){
$scope.quizzes = data;
//console.log(data);
});
};
$scope.getQuizByCampID = function (campid) {
alert(campid);
$http.post("js/ajax/getQuiz.php?campid="+campid).success(function(data){
$scope.quizzesById = data;
$scope.QuizInput = "";
});
};
$scope.addQuiz = function (quizid, quizname, campid) {
console.log(quizid + quizname + campid);
$http.post("js/ajax/addQuiz.php?quizid="+quizid+"&quizname="+quizname+"&campid="+campid).success(function(data){
getQuiz();
$scope.QuizInput = "";
});
};
}])
A:
You should store the value in a service.
example:
app.factory('SharedService', function() {
this.inputValue = null;
this.setInputValue = function(value) {
this.inputValue = value;
}
this.getInputValue = function() {
return this.inputValue;
}
return this;
});
Example on Plunkr
Read: AngularJS Docs on services
or check this Egghead.io video
|
Retroactive effects of irrelevant speech on serial recall from short-term memory.
The authors report 5 serial-recall experiments. In 4 of the 5 experiments, they show that irrelevant sound (IS) has a retroactive effect on material already in memory. In Experiment 1, IS presented during a filled retention interval had a reliable effect on list recall. Four further experiments, 3 of which used retroactive IS, showed that IS continued to-have an effect on recall following a long, filled retention interval. Articulatory suppression during visual input was found to abolish the long-lasting, retroactive effect of IS, supporting the idea that IS affects the phonological-loop component of short-term memory. IS also, therefore, seems to affect a longer term memory system with which the loop interacts. |
One hundred years of chronic obstructive pulmonary disease (COPD).
Chronic obstructive pulmonary disease (COPD) is an increasing health problem and one of the leading causes of morbidity and mortality worldwide, but knowledge about its pathogenesis has increased substantially in recent years. The disease results from interaction between individual risk factors (like enzymatic deficiencies) and environmental exposures to noxious agents, like cigarette smoking, occupational dusts, air pollution and infections in childhood. The main mechanisms that may contribute to airflow limitation in COPD are fixed narrowing of small airways, emphysema and luminal obstruction with mucus secretions. COPD is characterised by a chronic inflammatory process in the pulmonary tissue, with a pattern different from bronchial asthma, associated with extrapulmonary effects and is considered now a complex, systemic disease. Optimal therapeutic targeting of COPD depends on a clear understanding of the precise mechanisms of these complex processes and on early and correct evaluation of disease severity. A combination of pharmacological and non-pharmacological approaches is used to treat COPD. Bronchodilators are the mainstay of COPD treatment and can be combined with inhaled corticosteroids for greater efficacy and fewer side effects. The use of LTOT for hypoxemic patients has resulted in increased survival, and expanded drug therapy options have effectively improved dyspnoea and quality of life. Recent studies have documented the benefits of pulmonary rehabilitation. In addition, non-invasive mechanical ventilation offers new alternatives for patients with acute or chronic failure. |
The case of the vanished sword
By
washingtonpost.com editors
By John Lockwood
Washington
One of our memorials is missing a sword.
The General Sherman memorial just south of the Treasury Building shows General William Tecumseh Sherman on horseback atop a 32-foot pedestal guarded at each ground-level corner by a soldier. The memorial was designed by a Danish sculptor named Carl Rohl-Smith. It was dedicated on Oct. 15, 1903.
The northwest soldier is an infantryman, holding his rifle by the barrel, with the butt resting on the ground. The southwest soldier is an engineer, holding his rifle in the same position. He also carries a cylinder or tube, about 3 feet long. Perhaps it contains surveying instruments. The southeast soldier is a cavalryman, with sword pointed upward across his left shoulder. At the northeast is an artilleryman — whose hands close upon empty air.
When I saw the northeast soldier, the question arose: Is he missing a rifle or a sword? Well, what would one of my heroes, Sherlock Holmes, do? Look for a cartridge box, or a scabbard.
There was a scabbard there, an empty one. So it was indeed a missing sword — a fact later verified in The Post’s Oct. 16, 1903, edition, which included a drawing showing the soldier with a sword, its point touching the ground.
The lost sword is probably rusting in somebody’s attic or slowly corroding in a landfill. I doubt even Holmes could find it now. |
Q:
Pass values to IN operator in a Worklight SQL adapter
I have started to work with SQL adapters in Worklight, but I do not understand how can I pass values to an IN condition when invoking my adapter procedure.
A:
You will need to edit your question with your adapter's XML as well as implementation JavaScript...
Also, make sure to read the SQL adapters training module.
What you need to do is have your function get the values:
function myFunction (value1, value2) { ... }
And your SQL query will use them, like so (just as an example how to pass variables to any SQL query, doesn't matter if it contains an IN condition or not):
SELECT * FROM person where name='$[value1]' or id=$[value2];
Note the quotation marks for value1 (for text) and lack of for value2 (for numbers).
|
Bob Alcivar
Bob Alcivar (born July 8, 1938, in Chicago, Illinois) is an American music producer, composer, conductor and keyboard player. He is the father of rock keyboard player Jim Alcivar (Montrose, Gamma).
Discography
The Signatures - Their Voices and Instruments (1957) bass, arranger, vocals
The Signatures - Sing In (1958)
The Signatures - Prepare to Flip! (1959)
Julie London - Around Midnight (1960) - composer
The New Christy Minstrels - The Wandering Minstrels (1965) - vocal arrangement
The New Christy Minstrels - New Kick! (1966) arranger, director
The 5th Dimension - The Age of Aquarius (1969) - arranger
The Association - The Association (1969) - arranger
The Carnival - The Carnival (1969) - arranger
Seals & Crofts - Seals & Crofts (1970) - producer
The Sandpipers - Come Saturday Morning (1970) - producer & arranger
The 5th Dimension - Portrait (1970) - arranger
Sérgio Mendes & Brasil '77 - Love Music (1973) - arranger, keyboards, vocals
Tim Weisberg - Dreamspeaker - (1974) - arranger
Tom Waits - The Heart of Saturday Night (1974) - arranger
The 5th Dimension - Soul & Inspiration - (1974) - arranger
Sérgio Mendes & Brasil '77 - Vintage 74 - (1974) - vocal arrangement, rhythm arrangement
Sérgio Mendes & Brasil '77 - Sérgio Mendes - (1975) vocal arrangement
Montrose - Jump On It (1976) - string arrangement
Bette Midler - Broken Blossom - (1977) - arranger on "I Never Talk To Strangers" (duet with Tom Waits)
Bruce Johnston - Going Public (1977) - horn arrangement, string arrangement
Tim Weisberg - Live at Last (1977) - producer
Marilyn McCoo & Billy Davis, Jr. - The Two of Us (1977) - keyboards
Ronnie Montrose - Open Fire (1978) - orchestra arrangement, conductor
Tom Waits - Blue Valentine (1978) - orchestra
The Beach Boys - Keepin' the Summer Alive (1980) - horn arrangements
Tom Waits - Heartattack and Vine (1980) - string arrangement, orchestral arrangement, conductor
Seals & Crofts - Longest Road (1980) - string arrangement
Tom Waits - One from the Heart (1982) - piano, orchestral arrangement, conductor
Ceremony - Hang Out Your Poetry (1993) - arranger, string arrangement
Jazz at the Movies Band - One from the Heart: Sax at the Movies II (1994) - arranger, conductor
Royal Philharmonic Orchestra - Symphonic Sounds: The Music of Beach Boys (1998) - conductor, orchestral arrangement
Jazz at the Movies - The Bedroom Mixes (2000) - arranger
Bob Alcivar - Bahai Prayers - (2000)
Film
Butterflies Are Free (1972)
The Crazy World of Julius Vrooder (1974)
Olly Olly Oxen Free (1978)
One From the Heart (1982)
The Best Little Whorehouse in Texas (arranger, 1982)
Hysterical (1983)
That Secret Sunday (TV) (1986)
Blind Witness (TV) (1999)
Naked Lie [TV] (1989)
Roxanne: The Prize Pulitzer [TV] (1989)
Sparks: The Price of Passion [TV] (1990)
Deadly Medicine [TV] (1991)
External links
[ allmusic Biography]
Film Reference Biography
Category:1938 births
Category:Living people
Category:Musicians from Chicago
Category:20th-century American keyboardists
Category:Record producers from Illinois |
[Significance of serum CD62p and CD63 levels in patients with head injury].
To determine the serum levels of CD62p (alpha-granular membrane protein) and CD63 (lysosome intact membrane protein) in patients with head injury and to observe its relation to injury severity. Fifty-three patients with head injury were divided into 3 groups; Group A patients with mild head injury; Group B with moderate head injury; and Group C with severe head injury. The serum levels of CD62p, CD63 were measured on 12 h, d 1, 3, 5 and 7 after injury. The serum levels of CD62p and CD63 in Group B and Group C were higher than those in Group A and control (P<0.05). The serum level of CD62p in Group C was higher than that in Group B (P<0.05). The serum levels of CD62p in Group C on d 1, 3, 5 after injury were higher than those on 12 h (P<0.05). The serum level of CD63 in Group B on d 3 after injury were higher than that on 12 h (P<0.05). The serum levels of CD63 in Group C on d 1, 3, 5 after injury were higher than those on 12 h (P<0.05). The serum levels of CD62p and CD63 in patients with head injury may be helpful for identifying the severity of injury, and CD62p seems to be more sensitive. |
Peter Cooley
Peter Cooley (born November 19, 1940) is an American poet and Professor of English in the Department of English at Tulane University. He also directs Tulane's Creative Writing Program. Born in Detroit, Michigan, he holds degrees from Shimer College, the University of Chicago and the University of Iowa. He is the father of poet Nicole Cooley.
Career
Prior to joining Tulane, Cooley taught at the University of Wisconsin, Green Bay. He was the Robert Frost Fellow at the Bread Loaf Writers’ Conference in 1981.
Poetry and awards
Cooley has published several books of poetry with the Carnegie Mellon University Press.
He received the Inspirational Professor Award in 2001 and the Newcomb Professor of the Year Award in 2003. On August 14, 2015 he was named Louisiana's poet laureate.
Bibliography
Poetry
Collections
The Room Where Summer Ends (Pittsburgh: Carnegie Mellon University Press, 1979)
Nightseasons (Pittsburgh: Carnegie Mellon University Press, 1983)
The Van Gogh Notebook (Pittsburgh: Carnegie Mellon University Press, 1987)
The Astonished Hours (Pittsburgh: Carnegie Mellon University Press, 1992)
Sacred Conversations (Pittsburgh: Carnegie Mellon University Press, 1998)
A Place Made of Starlight (Pittsburgh: Carnegie Mellon University Press, 2003)
Divine Margins (Pittsburgh: Carnegie Mellon University Press, 2009)
Night Bus to the Afterlife (Pittsburgh: Carnegie Mellon University Press, 2014)
World Without Finishing (Pittsburgh: Carnegie Mellon University Press, 2018)
List of poems
References
External links
Peter Cooley listing in The Literary Encyclopedia
Peter Cooley’s faculty page, Tulane University
Peter Cooley author page at Virginia Quarterly Review, with links to poems
Category:1940 births
Category:Living people
Category:American male poets
Category:Poets Laureate of Louisiana
Category:Shimer College alumni
Category:The New Yorker people
Category:Tulane University faculty |
/**
* ScriptDev2 is an extension for mangos providing enhanced features for
* area triggers, creatures, game objects, instances, items, and spells beyond
* the default database scripting in mangos.
*
* Copyright (C) 2006-2013 ScriptDev2 <http://www.scriptdev2.com/>
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*
* World of Warcraft, and all World of Warcraft or Warcraft art, images,
* and lore are copyrighted by Blizzard Entertainment, Inc.
*/
/**
* ScriptData
* SDName: bug_trio
* SD%Complete: 75
* SDComment: Summon Player spell NYI; Poison Cloud damage spell NYI; Timers need adjustments
* SDCategory: Temple of Ahn'Qiraj
* EndScriptData
*/
#include "precompiled.h"
#include "temple_of_ahnqiraj.h"
enum
{
// kri
SPELL_CLEAVE = 26350,
SPELL_TOXIC_VOLLEY = 25812,
SPELL_SUMMON_CLOUD = 26590, // summons 15933
// vem
SPELL_CHARGE = 26561,
SPELL_VENGEANCE = 25790,
SPELL_KNOCKBACK = 26027,
// yauj
SPELL_HEAL = 25807,
SPELL_FEAR = 26580,
NPC_YAUJ_BROOD = 15621
};
struct MANGOS_DLL_DECL boss_kriAI : public ScriptedAI
{
boss_kriAI(Creature* pCreature) : ScriptedAI(pCreature)
{
m_pInstance = (ScriptedInstance*)pCreature->GetInstanceData();
Reset();
}
ScriptedInstance* m_pInstance;
uint32 m_uiCleaveTimer;
uint32 m_uiToxicVolleyTimer;
void Reset() override
{
m_uiCleaveTimer = urand(4000, 8000);
m_uiToxicVolleyTimer = urand(6000, 12000);
}
void JustDied(Unit* /*pKiller*/) override
{
// poison cloud on death
DoCastSpellIfCan(m_creature, SPELL_SUMMON_CLOUD, CAST_TRIGGERED);
if (!m_pInstance)
{
return;
}
// If the other 2 bugs are still alive, make unlootable
if (m_pInstance->GetData(TYPE_BUG_TRIO) != DONE)
{
m_creature->RemoveFlag(UNIT_DYNAMIC_FLAGS, UNIT_DYNFLAG_LOOTABLE);
m_pInstance->SetData(TYPE_BUG_TRIO, SPECIAL);
}
}
void JustReachedHome() override
{
if (m_pInstance)
{
m_pInstance->SetData(TYPE_BUG_TRIO, FAIL);
}
}
void UpdateAI(const uint32 uiDiff) override
{
// Return since we have no target
if (!m_creature->SelectHostileTarget() || !m_creature->getVictim())
{
return;
}
// Cleave_Timer
if (m_uiCleaveTimer < uiDiff)
{
if (DoCastSpellIfCan(m_creature->getVictim(), SPELL_CLEAVE) == CAST_OK)
{
m_uiCleaveTimer = urand(5000, 12000);
}
}
else
{ m_uiCleaveTimer -= uiDiff; }
// ToxicVolley_Timer
if (m_uiToxicVolleyTimer < uiDiff)
{
if (DoCastSpellIfCan(m_creature, SPELL_TOXIC_VOLLEY) == CAST_OK)
{
m_uiToxicVolleyTimer = urand(10000, 15000);
}
}
else
{ m_uiToxicVolleyTimer -= uiDiff; }
DoMeleeAttackIfReady();
}
};
struct MANGOS_DLL_DECL boss_vemAI : public ScriptedAI
{
boss_vemAI(Creature* pCreature) : ScriptedAI(pCreature)
{
m_pInstance = (ScriptedInstance*)pCreature->GetInstanceData();
Reset();
}
ScriptedInstance* m_pInstance;
uint32 m_uiChargeTimer;
uint32 m_uiKnockBackTimer;
void Reset() override
{
m_uiChargeTimer = urand(15000, 27000);
m_uiKnockBackTimer = urand(8000, 20000);
}
void JustDied(Unit* /*pKiller*/) override
{
// Enrage the other bugs
DoCastSpellIfCan(m_creature, SPELL_VENGEANCE, CAST_TRIGGERED);
if (!m_pInstance)
{
return;
}
// If the other 2 bugs are still alive, make unlootable
if (m_pInstance->GetData(TYPE_BUG_TRIO) != DONE)
{
m_creature->RemoveFlag(UNIT_DYNAMIC_FLAGS, UNIT_DYNFLAG_LOOTABLE);
m_pInstance->SetData(TYPE_BUG_TRIO, SPECIAL);
}
}
void JustReachedHome() override
{
if (m_pInstance)
{
m_pInstance->SetData(TYPE_BUG_TRIO, FAIL);
}
}
void UpdateAI(const uint32 uiDiff) override
{
// Return since we have no target
if (!m_creature->SelectHostileTarget() || !m_creature->getVictim())
{
return;
}
// Charge_Timer
if (m_uiChargeTimer < uiDiff)
{
if (Unit* pTarget = m_creature->SelectAttackingTarget(ATTACKING_TARGET_RANDOM, 0))
{
if (DoCastSpellIfCan(pTarget, SPELL_CHARGE) == CAST_OK)
{
m_uiChargeTimer = urand(8000, 16000);
}
}
}
else
{ m_uiChargeTimer -= uiDiff; }
// KnockBack_Timer
if (m_uiKnockBackTimer < uiDiff)
{
if (DoCastSpellIfCan(m_creature, SPELL_KNOCKBACK) == CAST_OK)
{
if (m_creature->GetThreatManager().getThreat(m_creature->getVictim()))
{
m_creature->GetThreatManager().modifyThreatPercent(m_creature->getVictim(), -80);
}
m_uiKnockBackTimer = urand(15000, 25000);
}
}
else
{ m_uiKnockBackTimer -= uiDiff; }
DoMeleeAttackIfReady();
}
};
struct MANGOS_DLL_DECL boss_yaujAI : public ScriptedAI
{
boss_yaujAI(Creature* pCreature) : ScriptedAI(pCreature)
{
m_pInstance = (ScriptedInstance*)pCreature->GetInstanceData();
Reset();
}
ScriptedInstance* m_pInstance;
uint32 m_uiHealTimer;
uint32 m_uiFearTimer;
void Reset() override
{
m_uiHealTimer = urand(25000, 40000);
m_uiFearTimer = urand(12000, 24000);
}
void JustDied(Unit* /*Killer*/) override
{
// Spawn 10 yauj brood on death
float fX, fY, fZ;
for (int i = 0; i < 10; ++i)
{
m_creature->GetRandomPoint(m_creature->GetPositionX(), m_creature->GetPositionY(), m_creature->GetPositionZ(), 10.0f, fX, fY, fZ);
m_creature->SummonCreature(NPC_YAUJ_BROOD, fX, fY, fZ, 0.0f, TEMPSUMMON_TIMED_OOC_DESPAWN, 30000);
}
if (!m_pInstance)
{
return;
}
// If the other 2 bugs are still alive, make unlootable
if (m_pInstance->GetData(TYPE_BUG_TRIO) != DONE)
{
m_creature->RemoveFlag(UNIT_DYNAMIC_FLAGS, UNIT_DYNFLAG_LOOTABLE);
m_pInstance->SetData(TYPE_BUG_TRIO, SPECIAL);
}
}
void JustReachedHome() override
{
if (m_pInstance)
{
m_pInstance->SetData(TYPE_BUG_TRIO, FAIL);
}
}
void UpdateAI(const uint32 uiDiff) override
{
// Return since we have no target
if (!m_creature->SelectHostileTarget() || !m_creature->getVictim())
{
return;
}
// Fear_Timer
if (m_uiFearTimer < uiDiff)
{
if (DoCastSpellIfCan(m_creature, SPELL_FEAR) == CAST_OK)
{
DoResetThreat();
m_uiFearTimer = 20000;
}
}
else
{ m_uiFearTimer -= uiDiff; }
// Heal
if (m_uiHealTimer < uiDiff)
{
if (Unit* pTarget = DoSelectLowestHpFriendly(100.0f))
{
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{
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else
{ m_uiHealTimer -= uiDiff; }
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};
CreatureAI* GetAI_boss_yauj(Creature* pCreature)
{
return new boss_yaujAI(pCreature);
}
CreatureAI* GetAI_boss_vem(Creature* pCreature)
{
return new boss_vemAI(pCreature);
}
CreatureAI* GetAI_boss_kri(Creature* pCreature)
{
return new boss_kriAI(pCreature);
}
void AddSC_bug_trio()
{
Script* pNewScript;
pNewScript = new Script;
pNewScript->Name = "boss_kri";
pNewScript->GetAI = &GetAI_boss_kri;
pNewScript->RegisterSelf();
pNewScript = new Script;
pNewScript->Name = "boss_vem";
pNewScript->GetAI = &GetAI_boss_vem;
pNewScript->RegisterSelf();
pNewScript = new Script;
pNewScript->Name = "boss_yauj";
pNewScript->GetAI = &GetAI_boss_yauj;
pNewScript->RegisterSelf();
}
|
---
abstract: 'The aim of this paper is to establish a global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a Lévy process and a Gaussian white noise experiment observed up to a time $T$, with $T$ tending to $\infty$. These approximations are given in the sense of the Le Cam distance, under some smoothness conditions on the unknown Lévy density. All the asymptotic equivalences are established by constructing explicit Markov kernels that can be used to reproduce one experiment from the other.'
address:
- '*Laboratoire LJK, Université Joseph Fourier UMR 5224 51, Rue des Mathématiques, Saint Martin d’Hères BP 53 38041 Grenoble Cedex 09*'
- 'Corresponding Author, [email protected]'
author:
- Ester Mariucci
bibliography:
- 'refs.bib'
title: Asymptotic equivalence for pure jump Lévy processes with unknown Lévy density and Gaussian white noise
---
Nonparametric experiments,Le Cam distance,asymptotic equivalence,Lévy processes. 62B15,(62G20,60G51).
Introduction
============
Lévy processes are a fundamental tool in modelling situations, like the dynamics of asset prices and weather measurements, where sudden changes in values may happen. For that reason they are widely employed, among many other fields, in mathematical finance. To name a simple example, the price of a commodity at time $t$ is commonly given as an exponential function of a Lévy process. In general, exponential Lévy models are proposed for their ability to take into account several empirical features observed in the returns of assets such as heavy tails, high-kurtosis and asymmetry (see [@tankov] for an introduction to financial applications).
From a mathematical point of view, Lévy processes are a natural extension of the Brownian motion which preserves the tractable statistical properties of its increments, while relaxing the continuity of paths. The jump dynamics of a Lévy process is dictated by its Lévy density, say $f$. If $f$ is continuous, its value at a point $x_0$ determines how frequent jumps of size close to $x_0$ are to occur per unit time. Concretely, if $X$ is a pure jump Lévy process with Lévy density $f$, then the function $f$ is such that $$\int_Af(x)dx=\frac{1}{t}{\ensuremath {\mathbb{E}}}\bigg[\sum_{s\leq t}{\ensuremath {\mathbb{I}}}_A(\Delta X_s)\bigg],$$ for any Borel set $A$ and $t>0$. Here, $\Delta X_s\equiv X_s-X_{s^-}$ denotes the magnitude of the jump of $X$ at time $s$ and ${\ensuremath {\mathbb{I}}}_A$ is the characteristic function. Thus, the Lévy measure $$\nu(A):=\int_A f(x)dx,$$ is the average number of jumps (per unit time) whose magnitudes fall in the set $A$. Understanding the jumps behavior, therefore requires to estimate the Lévy measure. Several recent works have treated this problem, see e.g. [@bel15] for an overview.
When the available data consists of the whole trajectory of the process during a time interval $[0,T]$, the problem of estimating $f$ may be reduced to estimating the intensity function of an inhomogeneous Poisson process (see, e.g. [@fig06; @rey03]). However, a continuous-time sampling is never available in practice and thus the relevant problem is that of estimating $f$ based on discrete sample data $X_{t_0},\dots,X_{t_n}$ during a time interval $[0,T_n]$. In that case, the jumps are latent (unobservable) variables and that clearly adds to the difficulty of the problem. From now on we will place ourselves in a high-frequency setting, that is we assume that the sampling interval $\Delta_n=t_i-t_{i-1}$ tends to zero as $n$ goes to infinity. Such a high-frequency based statistical approach has played a central role in the recent literature on nonparametric estimation for Lévy processes (see e.g. [@fig09; @comte10; @comte11; @bec12; @duval12]). Moreover, in order to make consistent estimation possible, we will also ask the observation time $T_n$ to tend to infinity in order to allow the identification of the jump part in the limit.
Our aim is to prove that, under suitable hypotheses, estimating the Lévy density $f$ is equivalent to estimating the drift of an adequate Gaussian white noise model. In general, asymptotic equivalence results for statistical experiments provide a deeper understanding of statistical problems and allow to single out their main features. The idea is to pass via asymptotic equivalence to another experiment which is easier to analyze. By definition, two sequences of experiments ${\ensuremath {\mathscr{P}}}_{1,n}$ and ${\ensuremath {\mathscr{P}}}_{2,n}$, defined on possibly different sample spaces, but with the same parameter set, are asymptotically equivalent if the Le Cam distance $\Delta({\ensuremath {\mathscr{P}}}_{1,n},{\ensuremath {\mathscr{P}}}_{2,n})$ tends to zero. For ${\ensuremath {\mathscr{P}}}_{i}=({\ensuremath {\mathscr{X}}}_i,{\ensuremath {\mathscr{A}}}_i, \big(P_{i,\theta}:\theta\in\Theta)\big)$, $i=1,2$, $\Delta({\ensuremath {\mathscr{P}}}_1,{\ensuremath {\mathscr{P}}}_2)$ is the symmetrization of the deficiency $\delta({\ensuremath {\mathscr{P}}}_1,{\ensuremath {\mathscr{P}}}_2)$ where $$\delta({\ensuremath {\mathscr{P}}}_{1},{\ensuremath {\mathscr{P}}}_{2})=\inf_K\sup_{\theta\in\Theta}\big\|KP_{1,\theta}-P_{2,\theta}\big\|_{TV}.$$ Here the infimum is taken over all randomizations from $({\ensuremath {\mathscr{X}}}_1,{\ensuremath {\mathscr{A}}}_1)$ to $({\ensuremath {\mathscr{X}}}_2,{\ensuremath {\mathscr{A}}}_2)$ and $\| \cdot \|_{TV}$ denotes the total variation distance. Roughly speaking, the Le Cam distance quantifies how much one fails to reconstruct (with the help of a randomization) a model from the other one and vice versa. Therefore, we say that $\Delta({\ensuremath {\mathscr{P}}}_1,{\ensuremath {\mathscr{P}}}_2)=0$ can be interpreted as “the models ${\ensuremath {\mathscr{P}}}_1$ and ${\ensuremath {\mathscr{P}}}_2$ contain the same amount of information about the parameter $\theta$.” The general definition of randomization is quite involved but, in the most frequent examples (namely when the sample spaces are Polish and the experiments dominated), it reduces to that of a Markov kernel. One of the most important feature of the Le Cam distance is that it can be also interpreted in terms of statistical decision theory (see [@lecam; @LC2000]; a short review is presented in the Appendix). As a consequence, saying that two statistical models are equivalent means that any statistical inference procedure can be transferred from one model to the other in such a way that the asymptotic risk remains the same, at least for bounded loss functions. Also, as soon as two models, ${\ensuremath {\mathscr{P}}}_{1,n}$ and ${\ensuremath {\mathscr{P}}}_{2,n}$, that share the same parameter space $\Theta$ are proved to be asymptotically equivalent, the same result automatically holds for the restrictions of both ${\ensuremath {\mathscr{P}}}_{1,n}$ and ${\ensuremath {\mathscr{P}}}_{2,n}$ to a smaller subclass of $\Theta$.
Historically, the first results of asymptotic equivalence in a nonparametric context date from 1996 and are due to [@BL] and [@N96]. The first two authors have shown the asymptotic equivalence of nonparametric regression and a Gaussian white noise model while the third one those of density estimation and white noise. Over the years many generalizations of these results have been proposed such as [@regression02; @GN2002; @ro04; @C2007; @cregression; @R2008; @C2009; @R2013; @schmidt14] for nonparametric regression or [@cmultinomial; @j03; @BC04] for nonparametric density estimation models. Another very active field of study is that of diffusion experiments. The first result of equivalence between diffusion models and Euler scheme was established in 1998, see [@NM]. In later papers generalizations of this result have been considered (see [@C14; @esterdiffusion]). Among others we can also cite equivalence results for generalized linear models [@GN], time series [@GN2006; @NM], diffusion models [@D; @CLN; @R2006; @rmultidimensionale], GARCH model [@B], functional linear regression [@M2011], spectral density estimation [@GN2010] and volatility estimation [@R11]. Negative results are somewhat harder to come by; the most notable among them are [@sam96; @B98; @wang02]. There is however a lack of equivalence results concerning processes with jumps. A first result in this sense is [@esterESAIM] in which global asymptotic equivalences between the experiments generated by the discrete or continuous observation of a path of a Lévy process and a Gaussian white noise experiment are established. More precisely, in that paper, we have shown that estimating the drift function $h$ from a continuously or discretely (high frequency) time inhomogeneous jump-diffusion process: $$\label{ch4X}
X_t=\int_0^th(s)ds+\int_0^t\sigma(s)dW_s +\sum_{i=1}^{N_t}Y_i,\quad t\in[0,T_n],$$ is asymptotically equivalent to estimate $h$ in the Gaussian model: $$ dy_t=h(t)dt+\sigma(t)dW_t, \quad t\in[0,T_n].$$
Here we try to push the analysis further and we focus on the case in which the considered parameter is the Lévy density and $X=(X_t)$ is a pure jump Lévy process (see [@carr02] for the interest of such a class of processes when modelling asset returns). More in details, we consider the problem of estimating the Lévy density (with respect to a fixed, possibly infinite, Lévy measure $\nu_0$ concentrated on $I\subseteq {\ensuremath {\mathbb{R}}}$) $f:=\frac{d\nu}{d\nu_0}:I\to {\ensuremath {\mathbb{R}}}$ from a continuously or discretely observed pure jump Lévy process $X$ with possibly infinite Lévy measure. Here $I\subseteq {\ensuremath {\mathbb{R}}}$ denotes a possibly infinite interval and $\nu_0$ is supposed to be absolutely continuous with respect to Lebesgue with a strictly positive density $g:=\frac{d\nu_0}{d{\ensuremath{\textnormal{Leb}}}}$. In the case where $\nu$ is of finite variation one may write: $$\label{eqn:ch4Levy}
X_t=\sum_{0<s\leq t}\Delta X_s$$ or, equivalently, $X$ has a characteristic function given by: $${\ensuremath {\mathbb{E}}}\big[e^{iuX_t}\big]=\exp\bigg(-t\bigg(\int_{I}(1-e^{iuy})\nu(dy)\bigg)\bigg).$$
We suppose that the function $f$ belongs to some a priori set ${\ensuremath {\mathscr{F}}}$, nonparametric in general. The discrete observations are of the form $X_{t_i}$, where $t_i=T_n\frac{i}{n}$, $i=0,\dots,n$ with $T_n=n\Delta_n\to \infty$ and $\Delta_n\to 0$ as $n$ goes to infinity. We will denote by ${\ensuremath {\mathscr{P}}}_n^{\nu_0}$ the statistical model associated with the continuous observation of a trajectory of $X$ until time $T_n$ (which is supposed to go to infinity as $n$ goes to infinity) and by ${\ensuremath {\mathscr{Q}}}_n^{\nu_0}$ the one associated with the observation of the discrete data $(X_{t_i})_{i=0}^n$. The aim of this paper is to prove that, under adequate hypotheses on ${\ensuremath {\mathscr{F}}}$ (for example, $f$ must be bounded away from zero and infinity; see Section \[subsec:ch4parameter\] for a complete definition), the models ${\ensuremath {\mathscr{P}}}_n^{\nu_0}$ and ${\ensuremath {\mathscr{Q}}}_n^{\nu_0}$ are both asymptotically equivalent to a sequence of Gaussian white noise models of the form: $$dy_t=\sqrt{f(t)}dt+\frac{1}{2\sqrt{T_n}}\frac{dW_t}{\sqrt{g(t)}},\quad t\in I.$$ As a corollary, we then get the asymptotic equivalence between ${\ensuremath {\mathscr{P}}}_n^{\nu_0}$ and ${\ensuremath {\mathscr{Q}}}_n^{\nu_0}$. The main results are precisely stated as Theorems \[ch4teo1\] and \[ch4teo2\]. A particular case of special interest arises when $X$ is a compound Poisson process, $\nu_0\equiv {\ensuremath{\textnormal{Leb}}}([0,1])$ and ${\ensuremath {\mathscr{F}}}\subseteq {\ensuremath {\mathscr{F}}}_{(\gamma,K,\kappa,M)}^I$ where, for fixed $\gamma\in (0,1]$ and $K,\kappa, M$ strictly positive constants, ${\ensuremath {\mathscr{F}}}_{(\gamma,K,\kappa,M)}^I$ is a class of continuously differentiable functions on $I$ defined as follows: $$\label{ch4:fholder}
{\ensuremath {\mathscr{F}}}_{(\gamma,K,\kappa,M)}^I=\Big\{f: \kappa\leq f(x)\leq M, \ |f'(x)-f'(y)|\leq K|x-y|^{\gamma},\ \forall x,y\in I\Big\}.$$ In this case, the statistical models ${\ensuremath {\mathscr{P}}}_n^{\nu_0}$ and ${\ensuremath {\mathscr{Q}}}_n^{\nu_0}$ are both equivalent to the Gaussian white noise model: $$dy_t=\sqrt{f(t)}dt+\frac{1}{2\sqrt{T_n}}dW_t,\quad t\in [0,1].$$ See Example \[ex:ch4CPP\] for more details. By a theorem of Brown and Low in [@BL], we obtain, a posteriori, an asymptotic equivalence with the regression model $$Y_i=\sqrt{f\Big(\frac{i}{T_n}\Big)}+\frac{1}{2\sqrt{T_n}}\xi_i, \quad \xi_i\sim{\ensuremath {\mathscr{Nn}}}(0,1), \quad i=1,\dots, [T_n].$$ Note that a similar form of a Gaussian shift was found to be asymptotically equivalent to a nonparametric density estimation experiment, see [@N96]. Let us mention that we also treat some explicit examples where $\nu_0$ is neither finite nor compactly-supported (see Examples \[ch4ex2\] and \[ex3\]). Without entering into any detail, we remark here that the methods are very different from those in [@esterESAIM]. In particular, since $f$ belongs to the discontinuous part of a Lévy process, rather then its continuous part, the Girsanov-type changes of measure are irrelevant here. We thus need new instruments, like the Esscher changes of measure.
Our proof is based on the construction, for any given Lévy measure $\nu$, of two adequate approximations $\hat \nu_m$ and $\bar \nu_m$ of $\nu$: the idea of discretizing the Lévy density already appeared in an earlier work with P. Étoré and S. Louhichi, [@etore13]. The present work is also inspired by the papers [@cmultinomial] (for a multinomial approximation), [@BC04] (for passing from independent Poisson variables to independent normal random variables) and [@esterESAIM] (for a Bernoulli approximation). This method allows us to construct explicit Markov kernels that lead from one model to the other; these may be applied in practice to transfer minimax estimators.
The paper is organized as follows: Sections \[subsec:ch4parameter\] and \[subsec:ch4experiments\] are devoted to make the parameter space and the considered statistical experiments precise. The main results are given in Section \[subsec:ch4mainresults\], followed by Section \[sec:ch4experiments\] in which some examples can be found. The proofs are postponed to Section \[sec:ch4proofs\]. The paper includes an Appendix recalling the definition and some useful properties of the Le Cam distance as well as of Lévy processes.
Assumptions and main results
============================
The parameter space {#subsec:ch4parameter}
-------------------
Consider a (possibly infinite) Lévy measure $\nu_0$ concentrated on a possibly infinite interval $I\subseteq{\ensuremath {\mathbb{R}}}$, admitting a density $g>0$ with respect to Lebesgue. The parameter space of the experiments we are concerned with is a class of functions ${\ensuremath {\mathscr{F}}}={\ensuremath {\mathscr{F}}}^{\nu_0,I}$ defined on $I$ that form a class of Lévy densities with respect to $\nu_0$: For each $f\in{\ensuremath {\mathscr{F}}}$, let $\nu$ (resp. $\hat \nu_m$) be the Lévy measure having $f$ (resp. $\hat f_m$) as a density with respect to $\nu_0$ where, for every $f\in{\ensuremath {\mathscr{F}}}$, $\hat f_m(x)$ is defined as follows.
Suppose first $x>0$. Given a positive integer depending on $n$, $m=m_n$, let $J_j:=(v_{j-1},v_j]$ where $v_1=\varepsilon_m\geq 0$ and $v_j$ are chosen in such a way that $$\label{eq:ch4Jj}
\mu_m:=\nu_0(J_j)=\frac{\nu_0\big((I\setminus[0,\varepsilon_m])\cap {\ensuremath {\mathbb{R}}}_+\big)}{m-1},\quad \forall j=2,\dots,m.$$ In the sequel, for the sake of brevity, we will only write $m$ without making explicit the dependence on $n$. Define $x_j^*:=\frac{\int_{J_j}x\nu_0(dx)}{\mu_m}$ and introduce a sequence of functions $0\leq V_j\leq \frac{1}{\mu_m}$, $j=2,\dots,m$ supported on $[x_{j-1}^*, x_{j+1}^*]$ if $j=3,\dots,m-1$, on $[\varepsilon_m, x_3^*]$ if $j=2$ and on $(I\setminus [0,x_{m-1}^*])\cap {\ensuremath {\mathbb{R}}}_+$ if $j=m$. The $V_j$’s are defined recursively in the following way.
- $V_2$ is equal to $\frac{1}{\mu_m}$ on the interval $(\varepsilon_m, x_2^*]$ and on the interval $(x_2^*,x_3^*]$ it is chosen so that it is continuous (in particular, $V_2(x_2^*)=\frac{1}{\mu_m}$), $\int_{x_2^*}^{x_3^*}V_2(y)\nu_0(dy)=\frac{\nu_0((x_2^*, v_2])}{\mu_m}$ and $V_2(x_3^*)=0$.
- For $j=3,\dots,m-1$ define $V_j$ as the function $\frac{1}{\mu_m}-V_{j-1}$ on the interval $[x_{j-1}^*,x_j^*]$. On $[x_j^*,x_{j+1}^*]$ choose $V_j$ continuous and such that $\int_{x_j^*}^{x_{j+1}^*}V_j(y)\nu_0(dy)=\frac{\nu_0((x_j^*,v_j])}{\mu_m}$ and $V_j(x_{j+1}^*)=0$.
- Finally, let $V_m$ be the function supported on $(I\setminus [0,x_{m-1}^*]) \cap {\ensuremath {\mathbb{R}}}_+$ such that $$\begin{aligned}
V_m(x)&=\frac{1}{\mu_m}-V_{m-1}(x), \quad\text{for } x \in [x_{m-1}^*,x_m^*],\\
V_m(x)&=\frac{1}{\mu_m}, \quad\text{for } x \in (I\setminus [0,x_m^*])\cap {\ensuremath {\mathbb{R}}}_+.\end{aligned}$$
(It is immediate to check that such a choice is always possible). Observe that, by construction, $$\sum_{j=2}^m V_j(x)\mu_m=1, \quad \forall x\in (I\setminus[0,\varepsilon_m])\cap {\ensuremath {\mathbb{R}}}_+ \quad \textnormal{and} \quad \int_{(I\setminus[0,\varepsilon_m])\cap {\ensuremath {\mathbb{R}}}_+}V_j(y)\nu_0(dy)=1.$$
Analogously, define $\mu_m^-=\frac{\nu_0\big((I\setminus[-\varepsilon_m,0])\cap {\ensuremath {\mathbb{R}}}_-\big)}{m-1}$ and $J_{-m},\dots,J_{-2}$ such that $\nu_0(J_{-j})=\mu_m^-$ for all $j$. Then, for $x<0$, $x_{-j}^*$ is defined as $x_j^*$ by using $J_{-j}$ and $\mu_m^-$ instead of $J_j$ and $\mu_m$ and the $V_{-j}$’s are defined with the same procedure as the $V_j$’s, starting from $V_{-2}$ and proceeding by induction.
Define $$\label{eq:ch4hatf}
\hat f_m(x)={\ensuremath {\mathbb{I}}}_{[-\varepsilon_m,\varepsilon_m]}(x)+\sum_{j=2}^m \bigg(V_j(x)\int_{J_j} f(y)\nu_0(dy)+V_{-j}(x)\int_{J_{-j}} f(y)\nu_0(dy)\bigg).$$ The definitions of the $V_j$’s above are modeled on the following example:
\[ex:Vj\] Let $\nu_0$ be the Lebesgue measure on $[0,1]$ and $\varepsilon_m=0$. Then $v_j=\frac{j-1}{m-1}$ and $x_j^*=\frac{2j-3}{2m-2}$, $j=2,\dots,m$. The standard choice for $V_j$ (based on the construction by [@cmultinomial]) is given by the piecewise linear functions interpolating the values in the points $x_j^*$ specified above:
The function $\hat f_m$ has been defined in such a way that the rate of convergence of the $L_2$ norm between the restriction of $f$ and $\hat f_m$ on $I\setminus[-\varepsilon_m,\varepsilon_m]$ is compatible with the rate of convergence of the other quantities appearing in the statements of Theorems \[ch4teo1\] and \[ch4teo2\]. For that reason, as in [@cmultinomial], we have not chosen a piecewise constant approximation of $f$ but an approximation that is, at least in the simplest cases, a piecewise linear approximation of $f$. Such a choice allows us to gain an order of magnitude on the convergence rate of $\|f-\hat f_m\|_{L_2(\nu_0|{I\setminus{[-\varepsilon_m,\varepsilon_m]}})}$ at least when ${\ensuremath {\mathscr{F}}}$ is a class of sufficiently smooth functions.
We now explain the assumptions we will need to make on the parameter $f \in {\ensuremath {\mathscr{F}}}= {\ensuremath {\mathscr{F}}}^{\nu_0, I}$. The superscripts $\nu_0$ and $I$ will be suppressed whenever this can lead to no confusion. We require that:
1. There exist constants $\kappa, M >0$ such that $\kappa\leq f(y)\leq M$, for all $y\in I$ and $f\in {\ensuremath {\mathscr{F}}}$.
For every integer $m=m_n$, we can consider $\widehat{\sqrt{f}}_m$, the approximation of $\sqrt{f}$ constructed as $\hat f_m$ above, i.e. $\widehat{\sqrt{f}}_m(x)=\displaystyle{{\ensuremath {\mathbb{I}}}_{[-\varepsilon_m,\varepsilon_m]}(x)+\sum_{\substack{j=-m\dots,m\\ j\neq -1,0,1.}}V_j(x)\int_{J_j} \sqrt{f(y)}\nu_0(dy)}$, and introduce the quantities: $$\begin{aligned}
A_m^2(f)&:= \int_{I\setminus \big[-\varepsilon_m,\varepsilon_m\big]}\Big(\widehat{\sqrt {f}}_m(y)-\sqrt{f(y)}\Big)^2\nu_0(dy),\\
B_m^2(f)&:= \sum_{\substack{j=-m\dots,m\\ j\neq -1,0,1.}}\bigg(\int_{J_j}\frac{\sqrt{f(y)}}{\sqrt{\nu_0(J_j)}}\nu_0(dy)-\sqrt{\nu(J_j)}\bigg)^2,\\
C_m^2(f)&:= \int_{-\varepsilon_m}^{\varepsilon_m}\big(\sqrt{f(t)}-1\big)^2\nu_0(dt).
\end{aligned}$$ The conditions defining the parameter space ${\ensuremath {\mathscr{F}}}$ are expressed by asking that the quantities introduced above converge quickly enough to zero. To state the assumptions of Theorem \[ch4teo1\] precisely, we will assume the existence of sequences of discretizations $m = m_n\to\infty$, of positive numbers $\varepsilon_m=\varepsilon_{m_n}\to 0$ and of functions $V_j$, $j = \pm 2, \dots, \pm m$, such that:
1. \[cond:ch4hellinger\] $\lim\limits_{n \to \infty}n\Delta_n\sup\limits_{f \in{\ensuremath {\mathscr{F}}}}\displaystyle{\int_{I\setminus(-\varepsilon_m,\varepsilon_m)}}\Big(f(x)-\hat f_m(x)\Big)^2 \nu_0(dx) = 0$.
2. \[cond:ch4ABC\]$\lim\limits_{n \to \infty}n\Delta_n\sup\limits_{f \in{\ensuremath {\mathscr{F}}}} \big(A_m^2(f)+B_m^2(f)+C_m^2(f)\big)=0$.
Remark in particular that Condition (C\[cond:ch4ABC\]) implies the following:
1. $\displaystyle \sup_{f\in{\ensuremath {\mathscr{F}}}}\int_I (\sqrt{f(y)}-1)^2 \nu_0(dy) \leq L,$
where $L = \sup_{f \in {\ensuremath {\mathscr{F}}}} \int_{-\varepsilon_m}^{\varepsilon_m} (\sqrt{f(x)}-1)^2\nu_0(dx) + (\sqrt{M}+1)^2\nu_0\big(I\setminus (-\varepsilon_m, \varepsilon_m)\big)$, for any choice of $m$ such that the quantity in the limit appearing in Condition (C\[cond:ch4ABC\]) is finite.
Theorem \[ch4teo2\] has slightly stronger hypotheses, defining possibly smaller parameter spaces: We will assume the existence of sequences $m_n$, $\varepsilon_m$ and $V_j$, $j = \pm 2, \dots, \pm m$ (possibly different from the ones above) such that Condition (C1) is verified and the following stronger version of Condition (C2) holds:
1. $\lim\limits_{n \to \infty}n\Delta_n\sup\limits_{f \in{\ensuremath {\mathscr{F}}}} \big(A_m^2(f)+B_m^2(f)+nC_m^2(f)\big)=0$.
Finally, some of our results have a more explicit statement under the hypothesis of finite variation which we state as:
- $\int_I (|x|\wedge 1)\nu_0(dx)<\infty$.
The Condition (C1) and those involving the quantities $A_m(f)$ and $B_m(f)$ all concern similar but slightly different approximations of $f$. In concrete examples, they may all be expected to have the same rate of convergence but to keep the greatest generality we preferred to state them separately. On the other hand, conditions on the quantity $C_m(f)$ are purely local around zero, requiring the parameters $f$ to converge quickly enough to 1.
\[ex:ch4esempi\] To get a grasp on Conditions (C1), (C2) we analyze here three different examples according to the different behavior of $\nu_0$ near $0\in I$. In all of these cases the parameter space ${\ensuremath {\mathscr{F}}}^{\nu_0, I}$ will be a subclass of ${\ensuremath {\mathscr{F}}}_{(\gamma,K,\kappa,M)}^I$ defined as in . Recall that the conditions (C1), (C2) and (C2’) depend on the choice of sequences $m_n$, $\varepsilon_m$ and functions $V_j$. For the first two of the three examples, where $I = [0,1]$, we will make the standard choice for $V_j$ of triangular and trapezoidal functions, similarly to those in Example \[ex:Vj\]. Namely, for $j = 3, \dots, m-1$ we have $$\label{eq:ch4vj}
V_j(x) = {\ensuremath {\mathbb{I}}}_{(x_{j-1}^*, x_j^*]}(x) \frac{x-x_{j-1}^*}{x_j^*-x_{j-1}^*} \frac{1}{\mu_m} + {\ensuremath {\mathbb{I}}}_{(x_{j}^*, x_{j+1}^*]}(x) \frac{x_{j+1}^*-x}{x_{j+1}^*-x_{j}^*} \frac{1}{\mu_m};$$ the two extremal functions $V_2$ and $V_m$ are chosen so that $V_2 \equiv \frac{1}{\mu_m}$ on $(\varepsilon_m, x_2^*]$ and $V_m \equiv \frac{1}{\mu_m}$ on $(x_m^*, 1]$. In the second example, where $\nu_0$ is infinite, one is forced to take $\varepsilon_m > 0$ and to keep in mind that the $x_j^*$ are not uniformly distributed on $[\varepsilon_m,1]$. Proofs of all the statements here can be found in Section \[subsec:esempi\].
**1. The finite case:** $\nu_0\equiv {\ensuremath{\textnormal{Leb}}}([0,1])$.
In this case we are free to choose ${\ensuremath {\mathscr{F}}}^{{\ensuremath{\textnormal{Leb}}}, [0,1]} = {\ensuremath {\mathscr{F}}}_{(\gamma, K, \kappa, M)}^{[0,1]}$. Indeed, as $\nu_0$ is finite, there is no need to single out the first interval $J_1=[0,\varepsilon_m]$, so that $C_m(f)$ does not enter in the proofs and the definitions of $A_m(f)$ and $B_m(f)$ involve integrals on the whole of $[0,1]$. Also, the choice of the $V_j$’s as in guarantees that $\int_0^1 V_j(x) dx = 1$. Then, the quantities $\|f-\hat f_m\|_{L_2([0,1])}$, $A_m(f)$ and $B_m(f)$ all have the same rate of convergence, which is given by: $$\sqrt{\int_0^1\Big(f(x)-\hat f_m(x)\Big)^2 \nu_0(dx)}+A_m(f)+B_m(f)=O\Big(m^{-\gamma-1}+m^{-\frac{3}{2}}\Big),$$ uniformly on $f$. See Section \[subsec:esempi\] for a proof.
**2. The finite variation case:** $\frac{d\nu_0}{d{\ensuremath{\textnormal{Leb}}}}(x)=x^{-1}{\ensuremath {\mathbb{I}}}_{[0,1]}(x)$.
In this case, the parameter space ${\ensuremath {\mathscr{F}}}^{\nu_0, [0,1]}$ is a proper subset of ${\ensuremath {\mathscr{F}}}_{(\gamma, K, \kappa, M)}^{[0,1]}$. Indeed, as we are obliged to choose $\varepsilon_m > 0$, we also need to impose that $C_m(f) = o\big(\frac{1}{n\sqrt{\Delta_n}}\big)$, with uniform constants with respect to $f$, that is, that all $f \in {\ensuremath {\mathscr{F}}}$ converge to 1 quickly enough as $x \to 0$. Choosing $\varepsilon_m = m^{-1-\alpha}$, $\alpha> 0$ we have that $\mu_m=\frac{\ln (\varepsilon_m^{-1})}{m-1}$, $v_j =\varepsilon_m^{\frac{m-j}{m-1}}$ and $x_j^* =\frac{(v_{j}-v_{j-1})}{\mu_m}$. In particular, $\max_j|v_{j-1}-v_j|=|v_m-v_{m-1}|=O\Big(\frac{\ln m}{m}\Big)$. Also in this case one can prove that the standard choice of $V_j$ described above leads to $\int_{\varepsilon_m}^1 V_j(x) \frac{dx}{x} = 1$. Again, the quantities $\|f-\hat f_m\|_{L_2(\nu_0|{I\setminus{[0,\varepsilon_m]}})}$, $A_m(f)$ and $B_m(f)$ have the same rate of convergence given by: $$\label{eq:ch4ex2}
\sqrt{\int_{\varepsilon_m}^1\Big(f(x)-\hat f_m(x)\Big)^2 \nu_0(dx)} +A_m(f)+B_m(f)=O\bigg(\bigg(\frac{\ln m}{m}\bigg)^{\gamma+1} \sqrt{\ln (\varepsilon_m^{-1})}\bigg),$$ uniformly on $f$. The condition on $C_m(f)$ depends on the behavior of $f$ near $0$. For example, it is ensured if one considers a parametric family of the form $f(x)=e^{-\lambda x}$ with a bounded $\lambda > 0$. See Section \[subsec:esempi\] for a proof.
**3. The infinite variation, non-compactly supported case:** $\frac{d\nu_0}{d{\ensuremath{\textnormal{Leb}}}}(x)=x^{-2}{\ensuremath {\mathbb{I}}}_{{\ensuremath {\mathbb{R}}}_+}(x)$.
This example involves significantly more computations than the preceding ones, since the classical triangular choice for the functions $V_j$ would not have integral equal to 1 (with respect to $\nu_0$), and the support is not compact. The parameter space ${\ensuremath {\mathscr{F}}}^{\nu_0, [0, \infty)}$ can still be chosen as a proper subclass of ${\ensuremath {\mathscr{F}}}_{(\gamma, K, \kappa, M)}^{[0,\infty)}$, again by imposing that $C_m(f)$ converges to zero quickly enough (more details about this condition are discussed in Example \[ex3\]). We divide the interval $[0, \infty)$ in $m$ intervals $J_j = [v_{j-1}, v_j)$ with: $$v_0 = 0; \quad v_1 = \varepsilon_m; \quad v_j = \frac{\varepsilon_m(m-1)}{m-j};\quad v_m = \infty; \quad \mu_m = \frac{1}{\varepsilon_m(m-1)}.$$ To deal with the non-compactness problem, we choose some “horizon” $H(m)$ that goes to infinity slowly enough as $m$ goes to infinity and we bound the $L_2$ distance between $f$ and $\hat f_m$ for $x > H(m)$ by $2\sup\limits_{x\geq H(m)}\frac{f(x)^2}{H(m)}$. We have: $$\|f-\hat f_m\|_{L_2(\nu_0|{I\setminus{[0,\varepsilon_m]}})}^2+A_m^2(f)+B_m^2(f)=O\bigg(\frac{H(m)^{3+4\gamma}}{(\varepsilon_m m)^{2+2\gamma}}+\sup_{x\geq H(m)}\frac{f(x)^2}{H(m)}\bigg).$$ In the general case where the best estimate for $\displaystyle{\sup_{x\geq H(m)}f(x)^2}$ is simply given by $M^2$, an optimal choice for $H(m)$ is $\sqrt{\varepsilon_m m}$, that gives a rate of convergence: $$\|f-\hat f_m\|_{L_2(\nu_0|{I\setminus{[0,\varepsilon_m]}})}^2+A_m^2(f)+B_m^2(f) =O\bigg( \frac{1}{\sqrt{\varepsilon_m m}}\bigg),$$ independently of $\gamma$. See Section \[subsec:esempi\] for a proof.
Definition of the experiments {#subsec:ch4experiments}
-----------------------------
Let $(x_t)_{t\geq 0}$ be the canonical process on the Skorokhod space $(D,{\ensuremath {\mathscr{D}}})$ and denote by $P^{(b,0,\nu)}$ the law induced on $(D,{\ensuremath {\mathscr{D}}})$ by a Lévy process with characteristic triplet $(b,0,\nu)$. We will write $P_t^{(b,0,\nu)}$ for the restriction of $P^{(b,0,\nu)}$ to the $\sigma$-algebra ${\ensuremath {\mathscr{D}}}_t$ generated by $\{x_s:0\leq s\leq t\}$ (see \[sec:ch4levy\] for the precise definitions). Let $Q_t^{(b,0,\nu)}$ be the marginal law at time $t$ of a Lévy process with characteristic triplet ${(b,0,\nu)}$. In the case where $\int_{|y|\leq 1}|y|\nu(dy)<\infty$ we introduce the notation $\gamma^{\nu}:=\int_{|y|\leq 1}y\nu(dy)$; then, Condition (H2) guarantees the finiteness of $\gamma^{\nu-\nu_0}$ (see Remark 33.3 in [@sato] for more details).
Recall that we introduced the discretization $t_i=T_n\frac{i}{n}$ of $[0,T_n]$ and denote by $\textbf Q_n^{(\gamma^{\nu-\nu_0},0,\nu)}$ the laws of the $n+1$ marginals of $(x_t)_{t\geq 0}$ at times $t_i$, $i=0,\dots,n$. We will consider the following statistical models, depending on a fixed, possibly infinite, Lévy measure $\nu_0$ concentrated on $I$ (clearly, the models with the subscript $FV$ are meaningful only under the assumption (FV)): $$\begin{aligned}
{\ensuremath {\mathscr{P}}}_{n,FV}^{\nu_0}&=\bigg(D,{\ensuremath {\mathscr{D}}}_{T_n},\Big\{P_{T_n}^{(\gamma^{\nu},0,\nu)}:f:=\frac{d\nu}{d\nu_0}\in{\ensuremath {\mathscr{F}}}^{\nu_0,I}\Big\}\bigg),\\
{\ensuremath {\mathscr{Q}}}_{n,FV}^{\nu_0}&=\bigg({\ensuremath {\mathbb{R}}}^{n+1},{\ensuremath {\mathscr{B}}}({\ensuremath {\mathbb{R}}}^{n+1}),\Big\{ \textbf Q_{n}^{(\gamma^{\nu},0,\nu)}:f:=\frac{d\nu}{d\nu_0}\in{\ensuremath {\mathscr{F}}}^{\nu_0,I}\Big\}\bigg),\\
{\ensuremath {\mathscr{P}}}_{n}^{\nu_0}&=\bigg(D,{\ensuremath {\mathscr{D}}}_{T_n},\Big\{P_{T_n}^{(\gamma^{\nu-\nu_0},0,\nu)}:f:=\frac{d\nu}{d\nu_0}\in{\ensuremath {\mathscr{F}}}^{\nu_0,I}\Big\}\bigg),\\
{\ensuremath {\mathscr{Q}}}_{n}^{\nu_0}&=\bigg({\ensuremath {\mathbb{R}}}^{n+1},{\ensuremath {\mathscr{B}}}({\ensuremath {\mathbb{R}}}^{n+1}),\Big\{\textbf Q_{n}^{(\gamma^{\nu-\nu_0},0,\nu)}:f:=\frac{d\nu}{d\nu_0}\in{\ensuremath {\mathscr{F}}}^{\nu_0,I}\Big\}\bigg).
\end{aligned}$$ Finally, let us introduce the Gaussian white noise model that will appear in the statement of our main results. For that, let us denote by $(C(I),{\ensuremath {\mathscr{C}}})$ the space of continuous mappings from $I$ into ${\ensuremath {\mathbb{R}}}$ endowed with its standard filtration, by $g$ the density of $\nu_0$ with respect to the Lebesgue measure. We will require $g>0$ and let $\mathbb W_n^f$ be the law induced on $(C(I),{\ensuremath {\mathscr{C}}})$ by the stochastic process satisfying: $$\begin{aligned}
\label{eqn:ch4Wf}
dy_t=\sqrt{f(t)}dt+\frac{dW_t}{2\sqrt{T_n}\sqrt{g(t)}}, \quad t\in I,\end{aligned}$$ where $(W_t)_{t\in{\ensuremath {\mathbb{R}}}}$ denotes a Brownian motion on ${\ensuremath {\mathbb{R}}}$ with $W_0=0$. Then we set: $${\ensuremath {\mathscr{W}}}_n^{\nu_0}=\Big(C(I),{\ensuremath {\mathscr{C}}},\{\mathbb W_n^{f}:f\in{\ensuremath {\mathscr{F}}}^{\nu_0,I}\}\Big).$$ Observe that when $\nu_0$ is a finite Lévy measure, then ${\ensuremath {\mathscr{W}}}_n^{\nu_0}$ is equivalent to the statistical model associated with the continuous observation of a process $(\tilde y_t)_{t\in I}$ defined by: $$\begin{aligned}
d\tilde y_t=\sqrt{f(t)g(t)}dt+\frac{d W_t}{2\sqrt{T_n}}, \quad t\in I.\end{aligned}$$
Main results {#subsec:ch4mainresults}
------------
Using the notation introduced in Section \[subsec:ch4parameter\], we now state our main results. For brevity of notation, we will denote by $H(f,\hat f_m)$ (resp. $L_2(f,\hat f_m)$) the Hellinger distance (resp. the $L_2$ distance) between the Lévy measures $\nu$ and $\hat\nu_m$ restricted to $I\setminus{[-\varepsilon_m,\varepsilon_m]}$, i.e.: $$\begin{aligned}
H^2(f,\hat f_m)&:=\int_{I\setminus{[-\varepsilon_m,\varepsilon_m]}}\Big(\sqrt{f(x)}-\sqrt{\hat f_m(x)}\Big)^2 \nu_0(dx),\\
L_2(f,\hat f_m)^2&:=\int_{I\setminus{[-\varepsilon_m,\varepsilon_m]}}\big(f(y)-\hat f_m(y)\big)^2\nu_0(dy).\end{aligned}$$ Observe that Condition (H1) implies (see Lemma \[lemma:ch4hellinger\]) $$\frac{1}{4M}L_2(f,\hat f_m)^2\leq H^2(f,\hat f_m)\leq \frac{1}{4\kappa}L_2(f,\hat f_m)^2.$$
\[ch4teo1\] Let $\nu_0$ be a known Lévy measure concentrated on a (possibly infinite) interval $I\subseteq {\ensuremath {\mathbb{R}}}$ and having strictly positive density with respect to the Lebesgue measure. Let us choose a parameter space ${\ensuremath {\mathscr{F}}}^{\nu_0, I}$ such that there exist a sequence $m = m_n$ of integers, functions $V_j$, $j = \pm 2, \dots, \pm m$ and a sequence $\varepsilon_m \to 0$ as $m \to \infty$ such that Conditions [(H1), (C1), (C2)]{.nodecor} are satisfied for ${\ensuremath {\mathscr{F}}}= {\ensuremath {\mathscr{F}}}^{\nu_0, I}$. Then, for $n$ big enough we have: $$\begin{aligned}
\Delta({\ensuremath {\mathscr{P}}}_n^{\nu_0}, {\ensuremath {\mathscr{W}}}_n^{\nu_0}) &= O\bigg(\sqrt{n\Delta_n}\sup_{f\in {\ensuremath {\mathscr{F}}}}\Big(A_m(f)+B_m(f)+C_m(f)\Big)\bigg) \nonumber \\
& +O\bigg(\sqrt{n\Delta_n}\sup_{f\in{\ensuremath {\mathscr{F}}}}L_2(f, \hat f_m)+\sqrt{\frac{m}{n\Delta_n}\Big(\frac{1}{\mu_m}+\frac{1}{\mu_m^-}\Big)}\bigg). \label{eq:teo1}\end{aligned}$$
\[ch4teo2\] Let $\nu_0$ be a known Lévy measure concentrated on a (possibly infinite) interval $I\subseteq {\ensuremath {\mathbb{R}}}$ and having strictly positive density with respect to the Lebesgue measure. Let us choose a parameter space ${\ensuremath {\mathscr{F}}}^{\nu_0, I}$ such that there exist a sequence $m = m_n$ of integers, functions $V_j$, $j = \pm 2, \dots, \pm m$ and a sequence $\varepsilon_m \to 0$ as $m \to \infty$ such that Conditions [(H1), (C1), (C2’)]{.nodecor} are satisfied for ${\ensuremath {\mathscr{F}}}= {\ensuremath {\mathscr{F}}}^{\nu_0, I}$. Then, for $n$ big enough we have: $$\begin{aligned}
\Delta({\ensuremath {\mathscr{Q}}}_n^{\nu_0}, {\ensuremath {\mathscr{W}}}_n^{\nu_0})& = O\bigg( \nu_0\Big(I\setminus[-\varepsilon_m,\varepsilon_m]\Big)\sqrt{n\Delta_n^2}+\frac{m\ln m}{\sqrt{n}}+\sqrt{n\sqrt{\Delta_n}\sup_{f\in{\ensuremath {\mathscr{F}}}}C_m(f)}\bigg) \nonumber \\
&+O\bigg(\sqrt{n\Delta_n}\sup_{f\in{\ensuremath {\mathscr{F}}}}\Big(A_m(f)+B_m(f)+H(f,\hat f_m)\Big)\bigg).\label{eq:teo2}\end{aligned}$$
\[cor:ch4generale\] Let $\nu_0$ be as above and let us choose a parameter space ${\ensuremath {\mathscr{F}}}^{\nu_0, I}$ so that there exist sequences $m_n'$, $\varepsilon_m'$, $V_j'$ and $m_n''$, $\varepsilon_m''$, $V_j''$ such that:
- Conditions (H1), (C1) and (C2) hold for $m_n'$, $\varepsilon_m'$, $V_j'$, and $\frac{m'}{n\Delta_n}\Big(\frac{1}{\mu_{m'}}+\frac{1}{\mu_{m'}^-}\Big)$ tends to zero.
- Conditions (H1), (C1) and (C2’) hold for $m_n''$, $\varepsilon_m''$, $V_j''$, and $\nu_0\Big(I\setminus[-\varepsilon_{m''},\varepsilon_{m''}]\Big)\sqrt{n\Delta_n^2}+\frac{m''\ln m''}{\sqrt{n}}$ tends to zero.
Then the statistical models ${\ensuremath {\mathscr{P}}}_{n}^{\nu_0}$ and ${\ensuremath {\mathscr{Q}}}_{n}^{\nu_0}$ are asymptotically equivalent: $$\lim_{n\to\infty}\Delta({\ensuremath {\mathscr{P}}}_{n}^{\nu_0},{\ensuremath {\mathscr{Q}}}_{n}^{\nu_0})=0,$$
If, in addition, the Lévy measures have finite variation, i.e. if we assume (FV), then the same results hold replacing ${\ensuremath {\mathscr{P}}}_{n}^{\nu_0}$ and ${\ensuremath {\mathscr{Q}}}_{n}^{\nu_0}$ by ${\ensuremath {\mathscr{P}}}_{n,FV}^{\nu_0}$ and ${\ensuremath {\mathscr{Q}}}_{n,FV}^{\nu_0}$, respectively (see Lemma \[ch4LC\]).
Examples {#sec:ch4experiments}
========
We will now analyze three different examples, underlining the different behaviors of the Lévy measure $\nu_0$ (respectively, finite, infinite with finite variation and infinite with infinite variation). The three chosen Lévy measures are ${\ensuremath {\mathbb{I}}}_{[0,1]}(x) dx$, ${\ensuremath {\mathbb{I}}}_{[0,1]}(x) \frac{dx}{x}$ and ${\ensuremath {\mathbb{I}}}_{{\ensuremath {\mathbb{R}}}_+}(x)\frac{dx}{x^2}$. In all three cases we assume the parameter $f$ to be uniformly bounded and with uniformly $\gamma$-Hölder derivatives: We will describe adequate subclasses ${\ensuremath {\mathscr{F}}}^{\nu_0, I} \subseteq {\ensuremath {\mathscr{F}}}_{(\gamma, K, \kappa, M)}^I$ defined as in . It seems very likely that the same results that are highlighted in these examples hold true for more general Lévy measures; however, we limit ourselves to these examples in order to be able to explicitly compute the quantities involved ($v_j$, $x_j^*$, etc.) and hence estimate the distance between $f$ and $\hat f_m$ as in Examples \[ex:ch4esempi\].
In the first of the three examples, where $\nu_0$ is the Lebesgue measure on $I=[0,1]$, we are considering the statistical models associated with the discrete and continuous observation of a compound Poisson process with Lévy density $f$. Observe that ${\ensuremath {\mathscr{W}}}_n^{{\ensuremath{\textnormal{Leb}}}}$ reduces to the statistical model associated with the continuous observation of a trajectory from: $$dy_t=\sqrt{f(t)}dt+\frac{1}{2\sqrt{T_n}}dW_t,\quad t\in [0,1].$$ In this case we have:
\[ex:ch4CPP\](Finite Lévy measure). Let $\nu_0$ be the Lebesgue measure on $I=[0,1]$ and let ${\ensuremath {\mathscr{F}}}= {\ensuremath {\mathscr{F}}}^{{\ensuremath{\textnormal{Leb}}}, [0,1]}$ be any subclass of ${\ensuremath {\mathscr{F}}}_{(\gamma, K, \kappa, M)}^{[0,1]}$ for some strictly positive constants $K$, $\kappa$, $M$ and $\gamma\in(0,1]$. Then: $$\lim_{n\to\infty}\Delta({\ensuremath {\mathscr{P}}}_{n,FV}^{{\ensuremath{\textnormal{Leb}}}},{\ensuremath {\mathscr{W}}}_n^{{\ensuremath{\textnormal{Leb}}}})=0 \ \textnormal{ and } \ \lim_{n\to\infty}\Delta({\ensuremath {\mathscr{Q}}}_{n,FV}^{{\ensuremath{\textnormal{Leb}}}},{\ensuremath {\mathscr{W}}}_n^{{\ensuremath{\textnormal{Leb}}}})=0.$$ More precisely, $$\Delta({\ensuremath {\mathscr{P}}}_{n,FV}^{{\ensuremath{\textnormal{Leb}}}},{\ensuremath {\mathscr{W}}}_n^{{\ensuremath{\textnormal{Leb}}}})=\begin{cases}O\Big((n\Delta_n)^{-\frac{\gamma}{4+2\gamma}}\Big)\quad \textnormal{if } \ \gamma\in\big(0,\frac{1}{2}\big],\\
O\Big((n \Delta_n)^{-\frac{1}{10}}\Big)\quad \textnormal{if } \ \gamma\in\big(\frac{1}{2},1\big].
\end{cases}$$ In the case where $\Delta_n = n^{-\beta}$, $\frac{1}{2} < \beta < 1$, an upper bound for the rate of convergence of $\Delta({\ensuremath {\mathscr{Q}}}_{n,FV}^{{\ensuremath{\textnormal{Leb}}}}, {\ensuremath {\mathscr{W}}}_n^{{\ensuremath{\textnormal{Leb}}}})$ is $$\Delta({\ensuremath {\mathscr{Q}}}_{n,FV}^{{\ensuremath{\textnormal{Leb}}}}, {\ensuremath {\mathscr{W}}}_n^{{\ensuremath{\textnormal{Leb}}}})=\begin{cases}
O\Big(n^{-\frac{\gamma+\beta}{4+2\gamma}}\ln n\Big)\quad \textnormal{if } \ \gamma\in\big(0,\frac{1}{2}\big) \text{ and }\frac{2+2\gamma}{3+2\gamma} \leq \beta < 1,\\
O\Big(n^{\frac{1}{2}-\beta}\ln n\Big)\quad \textnormal{if } \ \gamma\in\big(0,\frac{1}{2}\big) \text{ and } \frac{1}{2} < \beta < \frac{2+2\gamma}{3+2\gamma},\\
O\Big(n^{-\frac{2\beta+1}{10}}\ln n\Big)\quad \textnormal{if } \ \gamma\in\big[\frac{1}{2},1\big] \text{ and } \frac{3}{4} \leq \beta < 1,\\
O\Big(n^{\frac{1}{2}-\beta}\ln n\Big)\quad \textnormal{if } \ \gamma\in\big[\frac{1}{2},1\big] \text{ and } \frac{1}{2} < \beta < \frac{3}{4}.
\end{cases}$$ See Section \[subsec:ch4ex1\] for a proof.
\[ch4ex2\](Infinite Lévy measure with finite variation). Let $X$ be a truncated Gamma process with (infinite) Lévy measure of the form: $$\nu(A)=\int_A \frac{e^{-\lambda x}}{x}dx,\quad A\in{\ensuremath {\mathscr{B}}}([0,1]).$$ Here ${\ensuremath {\mathscr{F}}}^{\nu_0, I}$ is a 1-dimensional parametric family in $\lambda$, assuming that there exists a known constant $\lambda_0$ such that $0<\lambda\leq \lambda_0<\infty$, $f(t) = e^{-\lambda t}$ and $d\nu_0(x)=\frac{1}{x}dx$. In particular, the $f$ are Lipschitz, i.e. ${\ensuremath {\mathscr{F}}}^{\nu_0, [0,1]} \subset {\ensuremath {\mathscr{F}}}_{(\gamma = 1, K, \kappa, M)}^{[0,1]}$. The discrete or continuous observation (up to time $T_n$) of $X$ are asymptotically equivalent to ${\ensuremath {\mathscr{W}}}_n^{\nu_0}$, the statistical model associated with the observation of a trajectory of the process $(y_t)$: $$dy_t=\sqrt{f(t)}dt+\frac{\sqrt tdW_t}{2\sqrt{T_n}},\quad t\in[0,1].$$ More precisely, in the case where $\Delta_n = n^{-\beta}$, $\frac{1}{2} < \beta < 1$, an upper bound for the rate of convergence of $\Delta({\ensuremath {\mathscr{Q}}}_{n,FV}^{\nu_0}, {\ensuremath {\mathscr{W}}}_n^{\nu_0})$ is $$\Delta({\ensuremath {\mathscr{Q}}}_{n,FV}^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0}) = \begin{cases}
O\big(n^{\frac{1}{2}-\beta} \ln n\big) & \text{if } \frac{1}{2} < \beta \leq \frac{9}{10}\\
O\big(n^{-\frac{1+2\beta}{7}} \ln n\big) & \text{if } \frac{9}{10} < \beta < 1.
\end{cases}$$ Concerning the continuous setting we have: $$\Delta({\ensuremath {\mathscr{P}}}_{n,FV}^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0})=O\Big(n^{\frac{\beta-1}{6}} \big(\ln n\big)^{\frac{5}{2}}\Big) = O\Big(T_n^{-\frac{1}{6}} \big(\ln T_n\big)^\frac{5}{2}\Big).$$ See Section \[subsec:ch4ex2\] for a proof.
\[ex3\](Infinite Lévy measure, infinite variation). Let $X$ be a pure jump Lévy process with infinite Lévy measure of the form: $$\nu(A)=\int_A \frac{2-e^{-\lambda x^3}}{x^2}dx,\quad A\in{\ensuremath {\mathscr{B}}}({\ensuremath {\mathbb{R}}}^+).$$ Again, we are considering a parametric family in $\lambda > 0$, assuming that the parameter stays bounded below a known constant $\lambda_0$. Here, $f(t) =2- e^{-\lambda t^3}$, hence $1\leq f(t)\leq 2$, for all $t\geq 0$, and $f$ is Lipschitz, i.e. ${\ensuremath {\mathscr{F}}}^{\nu_0, {\ensuremath {\mathbb{R}}}_+} \subset {\ensuremath {\mathscr{F}}}_{(\gamma = 1, K, \kappa, M)}^{{\ensuremath {\mathbb{R}}}_+}$. The discrete or continuous observations (up to time $T_n$) of $X$ are asymptotically equivalent to the statistical model associated with the observation of a trajectory of the process $(y_t)$: $$dy_t=\sqrt{f(t)}dt+\frac{tdW_t}{2\sqrt{T_n}},\quad t\geq 0.$$ More precisely, in the case where $\Delta_n = n^{-\beta}$, $0 < \beta < 1$, an upper bound for the rate of convergence of $\Delta({\ensuremath {\mathscr{Q}}}_{n}^{\nu_0}, {\ensuremath {\mathscr{W}}}_n^{\nu_0})$ is $$\Delta({\ensuremath {\mathscr{Q}}}_{n}^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0}) = \begin{cases}
O\big(n^{\frac{1}{2} - \frac{2}{3}\beta}\big)& \text{if } \frac{3}{4} < \beta < \frac{12}{13}\\
O\big(n^{-\frac{1}{6}+\frac{\beta}{18}} (\ln n)^{\frac{7}{6}}\big) &\text{if } \frac{12}{13}\leq \beta<1.
\end{cases}$$
In the continuous setting, we have $$\Delta({\ensuremath {\mathscr{P}}}_{n}^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0})=O\big(n^{\frac{3\beta-3}{34}}(\ln n)^{\frac{7}{6}}\big) = O\big(T_n^{-\frac{3}{34}} (\ln T_n)^{\frac{7}{6}}\big).$$ See Section \[subsec:ch4ex3\] for a proof.
Proofs of the main results {#sec:ch4proofs}
==========================
In order to simplify notations, the proofs will be presented in the case $I\subseteq {\ensuremath {\mathbb{R}}}^+$. Nevertheless, this allows us to present all the main difficulties, since they can only appear near 0. To prove Theorems \[ch4teo1\] and \[ch4teo2\] we need to introduce several intermediate statistical models. In that regard, let us denote by $Q_j^f$ the law of a Poisson random variable with mean $T_n\nu(J_j)$ (see for the definition of $J_{j}$). We will denote by $\mathscr{L}_m$ the statistical model associated with the family of probabilities $\big\{\bigotimes_{j=2}^m Q_j^f:f\in{\ensuremath {\mathscr{F}}}\big\}$: $$\label{eq:ch4l}
\mathscr{L}_m=\bigg(\bar{{\ensuremath {\mathbb{N}}}}^{m-1},\mathcal P(\bar{{\ensuremath {\mathbb{N}}}}^{m-1}), \bigg\{\bigotimes_{j=2}^m Q_j^f:f\in{\ensuremath {\mathscr{F}}}\bigg\}\bigg).$$
By $N_{j}^f$ we mean the law of a Gaussian random variable ${\ensuremath {\mathscr{Nn}}}(2\sqrt{T_n\nu(J_j)},1)$ and by $\mathscr{N}_m$ the statistical model associated with the family of probabilities $\big\{\bigotimes_{j=2}^m N_j^f:f\in{\ensuremath {\mathscr{F}}}\big\}$: $$\label{eq:ch4n}
\mathscr{N}_m=\bigg({\ensuremath {\mathbb{R}}}^{m-1},\mathscr B({\ensuremath {\mathbb{R}}}^{m-1}), \bigg\{\bigotimes_{j=2}^m N_j^f:f\in{\ensuremath {\mathscr{F}}}\bigg\}\bigg).$$
For each $f\in{\ensuremath {\mathscr{F}}}$, let $\bar \nu_m$ be the measure having $\bar f_m$ as a density with respect to $\nu_0$ where, for every $f\in{\ensuremath {\mathscr{F}}}$, $\bar f_m$ is defined as follows. $$\label{eq:ch4barf}
\bar f_m(x):=
\begin{cases}
\quad 1 & \textnormal{if } x\in J_1,\\
\frac{\nu(J_j)}{{\nu_0}(J_{j})} & \textnormal{if } x\in J_{j}, \quad j = 2,\dots,m.
\end{cases}$$ Furthermore, define $$\label{eq:ch4modellobar}
\bar{\ensuremath {\mathscr{P}}}_{n}^{\nu_0}=\bigg(D,{\ensuremath {\mathscr{D}}}_{T_n},\Big\{P_{T_n}^{(\gamma^{\bar \nu_m-\nu_0},0,\bar\nu_m)}:\frac{d\bar\nu_m}{d\nu_0}\in{\ensuremath {\mathscr{F}}}\Big\}\bigg).$$
Proof of Theorem \[ch4teo1\]
----------------------------
We begin by a series of lemmas that will be needed in the proof. Before doing so, let us underline the scheme of the proof. We recall that the goal is to prove that estimating $f=\frac{d\nu}{d\nu_0}$ from the continuous observation of a Lévy process $(X_t)_{t\in[0,T_n]}$ without Gaussian part and having Lévy measure $\nu$ is asymptotically equivalent to estimating $f$ from the Gaussian white noise model:
$$dy_t=\sqrt{f(t)}dt+\frac{1}{2\sqrt{T_n g(t)}}dW_t,\quad g=\frac{d\nu_0}{d{\ensuremath{\textnormal{Leb}}}},\quad t\in I.$$
Also, recall the definition of $\hat \nu_m$ given in and read ${\ensuremath {\mathscr{P}}}_1 \overset{\Delta} \Longleftrightarrow {\ensuremath {\mathscr{P}}}_2$ as ${\ensuremath {\mathscr{P}}}_1$ is asymptotically equivalent to ${\ensuremath {\mathscr{P}}}_2$. Then, we can outline the proof in the following way.
- Step 1: $P_{T_n}^{(\gamma^{\nu-\nu_0},0,\nu)} \overset{\Delta} \Longleftrightarrow P_{T_n}^{(\gamma^{\hat\nu_m-\nu_0},0,\hat\nu_m)}$;
- Step 2: $P_{T_n}^{(\gamma^{\hat\nu_m-\nu_0},0,\hat\nu_m)} \overset{\Delta} \Longleftrightarrow \bigotimes_{j=2}^m {\ensuremath {\mathscr{P}}}(T_n\nu(J_j))$ (Poisson approximation).
Here $\bigotimes_{j=2}^m {\ensuremath {\mathscr{P}}}(T_n\nu(J_j))$ represents a statistical model associated with the observation of $m-1$ independent Poisson r.v. of parameters $T_n\nu(J_j)$;
- Step 3: $\bigotimes_{j=2}^m {\ensuremath {\mathscr{P}}}(T_n \nu(J_j)) \overset{\Delta} \Longleftrightarrow \bigotimes_{j=2}^m {\ensuremath {\mathscr{Nn}}}(2\sqrt{T_n\nu(J_j)},1)$ (Gaussian approximation);
- Step 4: $\bigotimes_{j=2}^m {\ensuremath {\mathscr{Nn}}}(2\sqrt{T_n\nu(J_j)},1)\overset{\Delta} \Longleftrightarrow (y_t)_{t\in I}$.
Lemmas \[lemma:ch4poisson\]–\[lemma:ch4kernel\], below, are the key ingredients of Step 2.
\[lemma:ch4poisson\] Let $\bar{\ensuremath {\mathscr{P}}}_{n}^{\nu_0}$ and $\mathscr{L}_m$ be the statistical models defined in and , respectively. Under the Assumption (H2) we have: $$\Delta(\bar{\ensuremath {\mathscr{P}}}_{n}^{\nu_0}, \mathscr{L}_m)=0, \textnormal{ for all } m.$$
Denote by $\bar {\ensuremath {\mathbb{N}}}={\ensuremath {\mathbb{N}}}\cup \{\infty\}$ and consider the statistics $S:(D,{\ensuremath {\mathscr{D}}}_{T_n})\to \big(\bar{\ensuremath {\mathbb{N}}}^{m-1},\mathcal{P}(\bar{\ensuremath {\mathbb{N}}}^{m-1})\big)$ defined by $$\label{eq:ch4S}
S(x)=\Big(N_{T_n}^{x;\,2},\dots,N_{T_n}^{x;\,m}\bigg)\quad \textnormal{with} \quad
N_{T_n}^{x;\,j}=\sum_{r\leq T_n}{\ensuremath {\mathbb{I}}}_{J_{j}}(\Delta x_r).$$ An application of Theorem \[ch4teosato\] to $P_{T_n}^{(\gamma^{\bar \nu_m-\nu_0},0,\bar \nu_m)}$ and $P_{T_n}^{(0,0,\nu_0)}$, yields $$\frac{d P_{T_n}^{(\gamma^{\bar \nu_m-\nu_0},0,\bar \nu_m)}}{dP_{T_n}^{(0,0,\nu_0)}}(x)=\exp\bigg(\sum_{j=2}^m \bigg(\ln\Big(\frac{\nu(J_j)}{\nu_0(J_j)}\Big)\bigg) N_{T_n}^{x;j}-T_n\int_I(\bar f_m(y)-1)\nu_0(dy)\bigg).$$ Hence, by means of the Fisher factorization theorem, we conclude that $S$ is a sufficient statistics for $\bar{\ensuremath {\mathscr{P}}}_{n}^{\nu_0}$. Furthermore, under $P_{T_n}^{(\gamma^{\bar \nu_m-\nu_0},0,\bar \nu_m)}$, the random variables $N_{T_n}^{x;j}$ have Poisson distributions $Q_{j}^f$ with means $T_n\nu(J_j)$. Then, by means of Property \[ch4fatto3\], we get $\Delta(\bar{\ensuremath {\mathscr{P}}}_{n}^{\nu_0}, \mathscr{L}_m)=0, \textnormal{ for all } m.$
Let us denote by $\hat Q_j^f$ the law of a Poisson random variable with mean $T_n\int_{J_j}\hat f_m(y)\nu_0(dy)$ and let $\hat{\mathscr{L}}_m$ be the statistical model associated with the family of probabilities $\{\bigotimes_{j=2}^m \hat Q_j^f:f\in {\ensuremath {\mathscr{F}}}\}$.
\[lemma:ch4poissonhatf\] $$\Delta(\mathscr L_m,\hat{\mathscr{L}}_m)\leq \sup_{f\in {\ensuremath {\mathscr{F}}}}\sqrt{\frac{T_n}{\kappa}\int_{I\setminus[0,\varepsilon_m]}\big(f(y)-\hat f_m(y)\big)^2\nu_0(dy)}.$$
By means of Facts \[ch4h\]–\[fact:ch4hellingerpoisson\], we get: $$\begin{aligned}
\Delta(\mathscr L_m,\hat{\mathscr{L}}_m)&\leq \sup_{f\in{\ensuremath {\mathscr{F}}}}H\bigg(\bigotimes_{j=2}^m Q_j^f,\bigotimes_{j=2}^m \hat Q_j^f\bigg)\\
&\leq \sup_{f\in{\ensuremath {\mathscr{F}}}}\sqrt{\sum_{j=2}^m 2 H^2(Q_j^f,\hat Q_j^f)}\\
& =\sup_{f\in{\ensuremath {\mathscr{F}}}}\sqrt 2\sqrt{\sum_{j=2}^m\bigg(1-\exp\bigg(-\frac{T_n}{2}\bigg[\sqrt{\int_{J_j}\hat f(y)\nu_0(dy)}-\sqrt{\int_{J_j} f(y)\nu_0(dy)}\bigg]^2\bigg)\bigg)}.\end{aligned}$$ By making use of the fact that $1-e^{-x}\leq x$ for all $x\geq 0$ and the equality $\sqrt a-\sqrt b= \frac{a-b}{\sqrt a+\sqrt b}$ combined with the lower bound $f\geq \kappa$ (that also implies $\hat f_m\geq \kappa$) and finally the Cauchy-Schwarz inequality, we obtain: $$\begin{aligned}
&1-\exp\bigg(-\frac{T_n}{2}\bigg[\sqrt{\int_{J_j}\hat f(y)\nu_0(dy)}-\sqrt{\int_{J_j} f(y)\nu_0(dy)}\bigg]^2\bigg)\\
&\leq \frac{T_n}{2}\bigg[\sqrt{\int_{J_j}\hat f(y)\nu_0(dy)}-\sqrt{\int_{J_j} f(y)\nu_0(dy)}\bigg]^2\\
& \leq \frac{T_n}{2} \frac{\bigg(\int_{J_j}(f(y)-\hat f_m(y))\nu_0(dy)\bigg)^2}{\kappa \nu_0(J_j)}\\
&\leq \frac{T_n}{2\kappa} \int_{J_j}\big(f(y)-\hat f_m(y)\big)^2\nu_0(dy).
\end{aligned}$$ Hence, $$H\bigg(\bigotimes_{j=2}^m Q_j^f,\bigotimes_{j=2}^m \hat Q_j^f\bigg)\leq \sqrt{\frac{T_n}{\kappa}\int_{I\setminus[0,\varepsilon_m]}\big(f(y)-\hat f_m(y)\big)^2\nu_0(dy)}.$$
\[lemma:ch4kernel\] Let $\hat\nu_m$ and $\bar \nu_m$ the Lévy measures defined as in and , respectively. For every $f\in {\ensuremath {\mathscr{F}}}$, there exists a Markov kernel $K$ such that $$KP_{T_n}^{(\gamma^{\bar\nu_m-\nu_0},0,\bar\nu_m)}=P_{T_n}^{(\gamma^{\hat \nu_m-\nu_0},0,\hat \nu_m)}.$$
By construction, $\bar\nu_m$ and $\hat\nu_m$ coincide on $[0,\varepsilon_m]$. Let us denote by $\bar \nu_m^{\textnormal{res}}$ and $\hat\nu_m^{\textnormal{res}}$ the restriction on $I\setminus[0,\varepsilon_m]$ of $\bar\nu_m$ and $\hat\nu_m$ respectively, then it is enough to prove: $KP_{T_n}^{(\gamma^{\bar\nu_m^{\textnormal{res}}-\nu_0},0,\bar\nu_m^{\textnormal{res}})}=P_{T_n}^{(\gamma^{\hat \nu_m^{\textnormal{res}}-\nu_0},0,\hat \nu_m^{\textnormal{res}})}.$ First of all, let us observe that the kernel $M$: $$M(x,A)=\sum_{j=2}^m{\ensuremath {\mathbb{I}}}_{J_j}(x)\int_A V_j(y)\nu_0(dy),\quad x\in I\setminus[0,\varepsilon_m],\quad A\in{\ensuremath {\mathscr{B}}}(I\setminus[0,\varepsilon_m])$$ is defined in such a way that $M \bar\nu_m^{\textnormal{res}} = \hat \nu_m^{\textnormal{res}}$. Indeed, for all $A\in{\ensuremath {\mathscr{B}}}(I\setminus[0,\varepsilon_m])$, $$\begin{aligned}
M\bar\nu_m^{\textnormal{res}}(A)&=\sum_{j=2}^m\int_{J_j}M(x,A)\bar\nu_m^{\textnormal{res}}(dx)=\sum_{j=2}^m \int_{J_j}\bigg(\int_A V_j(y)\nu_0(dy)\bigg)\bar\nu_m^{\textnormal{res}}(dx)\nonumber\\
&=\sum_{j=2}^m \bigg(\int_A V_j(y)\nu_0(dy)\bigg)\nu(J_j)=\int_A \hat f_m(y)\nu_0(dy)=\hat \nu_m^{\textnormal{res}}(A). \label{eqn:M}
\end{aligned}$$
Observe that $(\gamma^{\bar\nu_m^{\textnormal{res}}-\nu_0},0,\bar\nu_m^{\textnormal{res}})$ and $(\gamma^{\hat \nu_m^{\textnormal{res}}-\nu_0},0,\hat \nu_m^{\textnormal{res}})$ are Lévy triplets associated with compound Poisson processes since $\bar\nu_m^{\textnormal{res}}$ and $\hat \nu_m^{\textnormal{res}}$ are finite Lévy measures. The Markov kernel $K$ interchanging the laws of the Lévy processes is constructed explicitly in the case of compound Poisson processes. Indeed if $\bar X$ is the compound Poisson process having Lévy measure $\bar\nu_m^{\textnormal{res}}$, then $\bar X_{t} = \sum_{i=1}^{N_t} \bar Y_{i}$, where $N_t$ is a Poisson process of intensity $\iota_m:=\bar\nu_m^{\textnormal{res}}(I\setminus [0,\varepsilon_m])$ and the $\bar Y_{i}$ are i.i.d. random variables with probability law $\frac{1}{\iota_m}\bar\nu_m^{\textnormal{res}}$. Moreover, given a trajectory of $\bar X$, both the trajectory $(n_t)_{t\in[0,T_n]}$ of the Poisson process $(N_t)_{t\in[0,T_n]}$ and the realizations $\bar y_i$ of $\bar Y_i$, $i=1,\dots,n_{T_n}$ are uniquely determined. This allows us to construct $n_{T_n}$ i.i.d. random variables $\hat Y_i$ as follows: For every realization $\bar y_i$ of $\bar Y_i$, we define the realization $\hat y_i$ of $\hat Y_i$ by throwing it according to the probability law $M(\bar y_i,\cdot)$. Hence, thanks to , $(\hat Y_i)_i$ are i.i.d. random variables with probability law $\frac{1}{\iota_m} \hat \nu_m^{\text{res}}$. The desired Markov kernel $K$ (defined on the Skorokhod space) is then given by: $$K : (\bar X_{t})_{t\in[0,T_n]} \longmapsto \bigg(\hat X_{t} := \sum_{i=1}^{N_t} \hat Y_{i}\bigg)_{t\in[0,T_n]}.$$ Finally, observe that, since $$\begin{aligned}
\iota_m=\int_{I\setminus[0,\varepsilon_m]}\bar f_m(y)\nu_0(dy)&=\int_{I\setminus[0,\varepsilon_m]} f(y)\nu_0(dy)=\int_{I\setminus[0,\varepsilon_m]}\hat f_m(y)\nu_0(dy),
\end{aligned}$$ $(\hat X_t)_{t\in[0,T_n]}$ is a compound Poisson process with Lévy measure $\hat\nu_m^{\textnormal{res}}.$
Let us now state two lemmas needed to understand Step 4.
\[lemma:ch4wn\] Denote by ${\ensuremath {\mathscr{W}}}_m^\#$ the statistical model associated with the continuous observation of a trajectory from the Gaussian white noise: $$dy_t=\sqrt{f(t)}dt+\frac{1}{2\sqrt{T_n}\sqrt{g(t)}}dW_t,\quad t\in I\setminus [0,\varepsilon_m].$$ Then, according with the notation introduced in Section \[subsec:ch4parameter\] and at the beginning of Section \[sec:ch4proofs\], we have $$\Delta(\mathscr{N}_m,{\ensuremath {\mathscr{W}}}_m^\#)\leq 2\sqrt{T_n}\sup_{f\in {\ensuremath {\mathscr{F}}}} \big(A_m(f)+B_m(f)\big).$$
As a preliminary remark observe that ${\ensuremath {\mathscr{W}}}_m^\#$ is equivalent to the model that observes a trajectory from: $$d\bar y_t=\sqrt{f(t)}g(t)dt+\frac{\sqrt{g(t)}}{2\sqrt{T_n}}dW_t,\quad t\in I\setminus [0,\varepsilon_m].$$ Let us denote by $\bar Y_j$ the increments of the process $(\bar y_t)$ over the intervals $J_j$, $j=2,\dots,m$, i.e. $$\bar Y_j:=\bar y_{v_j}-\bar y_{v_{j-1}}\sim{\ensuremath {\mathscr{Nn}}}\bigg(\int_{J_j}\sqrt{f(y)}\nu_0(dy),\frac{\nu_0(J_j)}{4T_n}\bigg)$$ and denote by $\bar{\mathscr{N}}_m$ the statistical model associated with the distributions of these increments. As an intermediate result, we will prove that $$\label{eq:ch4normali}
\Delta(\mathscr{N}_m,\bar{\mathscr{N}}_m)\leq 2\sqrt{T_n} \sup_{f\in {\ensuremath {\mathscr{F}}}} B_m(f), \ \textnormal{ for all m}.$$ To that aim, remark that the experiment $\bar{\mathscr{N}}_m$ is equivalent to observing $m-1$ independent Gaussian random variables of means $\frac{2\sqrt{T_n}}{\sqrt{\nu_0(J_j)}}\int_{J_j}\sqrt{f(y)}\nu_0(dy)$, $j=2,\dots,m$ and variances identically $1$, name this last experiment $\mathscr{N}^{\#}_m$. Hence, using also Property \[ch4delta0\], Facts \[ch4h\] and \[fact:ch4gaussiane\] we get: $$\begin{aligned}
\Delta(\mathscr{N}_m, \bar{\mathscr{N}}_m)\leq\Delta(\mathscr{N}_m, \mathscr{N}^{\#}_m)&\leq \sqrt{\sum_{j=2}^m\bigg(\frac{2\sqrt{T_n}}{\sqrt{\nu_0(J_j)}}\int_{J_j}\sqrt{f(y)}\nu_0(dy)-2\sqrt{T_n\nu(J_j)}\bigg)^2}.\end{aligned}$$ Since it is clear that $\delta({\ensuremath {\mathscr{W}}}_m^\#,\bar{\mathscr{N}}_m)=0$, in order to bound $\Delta(\mathscr{N}_m,{\ensuremath {\mathscr{W}}}_m^\#)$ it is enough to bound $\delta(\bar{\mathscr{N}}_m,{\ensuremath {\mathscr{W}}}_m^\#)$. Using similar ideas as in [@cmultinomial] Section 8.2, we define a new stochastic process as: $$Y_t^*=\sum_{j=2}^m\bar Y_j\int_{\varepsilon_m}^t V_j(y)\nu_0(dy)+\frac{1}{2\sqrt{T_n}}\sum_{j=2}^m\sqrt{\nu_0(J_j)}B_j(t),\quad t\in I\setminus [0,\varepsilon_m],$$ where the $(B_j(t))$ are independent centered Gaussian processes independent of $(W_t)$ and with variances $$\textnormal{Var}(B_j(t))=\int_{\varepsilon_m}^tV_j(y)\nu_0(dy)-\bigg(\int_{\varepsilon_m}^tV_j(y)\nu_0(dy)\bigg)^2.$$ These processes can be constructed from a standard Brownian bridge $\{B(s), s\in[0,1]\}$, independent of $(W_t)$, via $$B_i(t)=B\bigg(\int_{\varepsilon_m}^t V_i(y)\nu_0(dy)\bigg).$$ By construction, $(Y_t^*)$ is a Gaussian process with mean and variance given by, respectively: $$\begin{aligned}
{\ensuremath {\mathbb{E}}}[Y_t^*]&=\sum_{j=2}^m{\ensuremath {\mathbb{E}}}[\bar Y_j]\int_{\varepsilon_m}^t V_j(y)\nu_0(dy)=\sum_{j=2}^m\bigg(\int_{J_j}\sqrt{f(y)}\nu_0(dy)\bigg)\int_{\varepsilon_m}^t V_j(y)\nu_0(dy),\\
\textnormal{Var}[Y_t^*]&=\sum_{j=2}^m\textnormal{Var}[\bar Y_j]\bigg(\int_{\varepsilon_m}^t V_j(y)\nu_0(dy)\bigg)^2+\frac{1}{4T_n}\sum_{j=2}^m \nu_0(J_j)\textnormal{Var}(B_j(t))\\
&= \frac{1}{4T_n}\int_{\varepsilon_m}^t \sum_{j=2}^m \nu_0(J_j) V_j(y)\nu_0(dy)= \frac{1}{4T_n}\int_{\varepsilon_m}^t \nu_0(dy)=\frac{\nu_0([\varepsilon_m,t])}{4T_n}.\end{aligned}$$ One can compute in the same way the covariance of $(Y_t^*)$ finding that $$\textnormal{Cov}(Y_s^*,Y_t^*)=\frac{\nu_0([\varepsilon_m,s])}{4 T_n}, \ \forall s\leq t.$$ We can then deduce that $$Y^*_t=\int_{\varepsilon_m}^t \widehat{\sqrt {f}}_m(y)\nu_0(dy)+\int_{\varepsilon_m}^t\frac{\sqrt{g(s)}}{2\sqrt{T_n}}dW^*_s,\quad t\in I\setminus [0,\varepsilon_m],$$ where $(W_t^*)$ is a standard Brownian motion and $$\widehat{\sqrt {f}}_m(x):=\sum_{j=2}^m\bigg(\int_{J_j}\sqrt{f(y)}\nu_0(dy)\bigg)V_j(x).$$
Applying Fact \[fact:ch4processigaussiani\], we get that the total variation distance between the process $(Y_t^*)_{t\in I\setminus [0,\varepsilon_m]}$ constructed from the random variables $\bar Y_j$, $j=2,\dots,m$ and the Gaussian process $(\bar y_t)_{t\in I\setminus [0,\varepsilon_m]}$ is bounded by $$\sqrt{4 T_n\int_{I\setminus [0,\varepsilon_m]}\big(\widehat{\sqrt {f}}_m-\sqrt{f(y)}\big)^2\nu_0(dy)},$$ which gives the term in $A_m(f)$.
\[lemma:ch4limitewn\] In accordance with the notation of Lemma \[lemma:ch4wn\], we have: $$\label{eq:ch4wn}
\Delta({\ensuremath {\mathscr{W}}}_m^\#,{\ensuremath {\mathscr{W}}}_n^{\nu_0})=O\bigg(\sup_{f\in{\ensuremath {\mathscr{F}}}}\sqrt{T_n\int_0^{\varepsilon_m}\big(\sqrt{f(t)}-1\big)^2\nu_0(dt)}\bigg).$$
Clearly $\delta({\ensuremath {\mathscr{W}}}_n^{\nu_0},{\ensuremath {\mathscr{W}}}_m^\#)=0$. To show that $\delta({\ensuremath {\mathscr{W}}}_m^\#,{\ensuremath {\mathscr{W}}}_n^{\nu_0})\to 0$, let us consider a Markov kernel $K^\#$ from $C(I\setminus [0,\varepsilon_m])$ to $C(I)$ defined as follows: Introduce a Gaussian process, $(B_t^m)_{t\in[0,\varepsilon_m]}$ with mean equal to $t$ and covariance $$\textnormal{Cov}(B_s^m,B_t^m)=\int_0^{\varepsilon_m}\frac{1}{4 T_n g(s)}{\ensuremath {\mathbb{I}}}_{[0,s]\cap [0,t]}(z)dz.$$ In particular, $$\textnormal{Var}(B_t^m)=\int_0^t\frac{1}{4 T_n g(s)}ds.$$ Consider it as a process on the whole of $I$ by defining $B_t^m=B_{\varepsilon_m}^m$ $\forall t>\varepsilon_m$. Let $\omega_t$ be a trajectory in $C(I\setminus [0,\varepsilon_m])$, which again we constantly extend to a trajectory on the whole of $I$. Then, we define $K^\#$ by sending the trajectory $\omega_t$ to the trajectory $\omega_t + B_t^m$. If we define $\mathbb{\tilde W}_n$ as the law induced on $C(I)$ by $$d\tilde{y}_t = h(t) dt + \frac{dW_t}{2\sqrt{T_n g(t)}}, \quad t \in I,\quad h(t) = \begin{cases}
1 & t \in [0, \varepsilon_m]\\
\sqrt{f(t)} & t \in I\setminus [0,\varepsilon_m],
\end{cases}$$ then $K^\# \mathbb{W}_n^f|_{I\setminus [0,\varepsilon_m]} = \mathbb{\tilde W}_n$, where $\mathbb{W}_n^f$ is defined as in . By means of Fact \[fact:ch4processigaussiani\] we deduce .
The proof of the theorem follows by combining the previous lemmas together:
- Step 1: Let us denote by $\hat{\ensuremath {\mathscr{P}}}_{n,m}^{\nu_0}$ the statistical model associated with the family of probabilities $(P_{T_n}^{(\gamma^{\hat\nu_m-\nu_0},0,\hat\nu_m)}:\frac{d\nu}{d\nu_0}\in{\ensuremath {\mathscr{F}}})$. Thanks to Property \[ch4delta0\], Fact \[ch4h\] and Theorem \[teo:ch4bound\] we have that $$\Delta({\ensuremath {\mathscr{P}}}_n^{\nu_0},\hat{\ensuremath {\mathscr{P}}}_{n,m}^{\nu_0})\leq \sqrt{\frac{T_n}{2}}\sup_{f\in {\ensuremath {\mathscr{F}}}}H(f,\hat f_m).$$
- Step 2: On the one hand, thanks to Lemma \[lemma:ch4poisson\], one has that the statistical model associated with the family of probability $(P_{T_n}^{(\gamma^{\bar \nu_m-\nu_0},0,\bar\nu_m)}:\frac{d\nu}{d\nu_0}\in{\ensuremath {\mathscr{F}}})$ is equivalent to $\mathscr{L}_m$. By means of Lemma \[lemma:ch4poissonhatf\] we can bound $\Delta(\mathscr{L}_m,\hat{\mathscr{L}}_m)$. On the other hand it is easy to see that $\delta(\hat{\ensuremath {\mathscr{P}}}_{n,m}^{\nu_0}, \hat{\mathscr{L}}_m)=0$. Indeed, it is enough to consider the statistics $$S: x \mapsto \bigg(\sum_{r\leq T_n}{\ensuremath {\mathbb{I}}}_{J_2}(\Delta x_r),\dots,\sum_{r\leq T_n}{\ensuremath {\mathbb{I}}}_{J_m}(\Delta x_r)\bigg)$$ since the law of the random variable $\sum_{r\leq T_n}{\ensuremath {\mathbb{I}}}_{J_j}(\Delta x_r)$ under $P_{T_n}^{(\gamma^{\hat\nu_m-\nu_0},0,\hat\nu_m)}$ is Poisson of parameter $T_n\int_{J_j}\hat f_m(y)\nu_0(dy)$ for all $j=2,\dots,m$. Finally, Lemmas \[lemma:ch4poisson\] and \[lemma:ch4kernel\] allows us to conclude that $\delta(\mathscr{L}_m,\hat{\ensuremath {\mathscr{P}}}_{n,m}^{\nu_0})=0$. Collecting all the pieces together, we get $$\Delta(\hat{\ensuremath {\mathscr{P}}}_{n,m}^{\nu_0},\mathscr{L}_m)\leq \sup_{f\in {\ensuremath {\mathscr{F}}}}\sqrt{\frac{T_n}{\kappa}\int_{I\setminus[0,\varepsilon_m]}\big(f(y)-\hat f_m(y)\big)^2\nu_0(dy)}.$$
- Step 3: Applying Theorem \[ch4teomisto\] and Fact \[ch4hp\] we can pass from the Poisson approximation given by $\mathscr{L}_m$ to a Gaussian one obtaining $$\Delta(\mathscr{L}_m,\mathscr{N}_m)=C\sup_{f\in {\ensuremath {\mathscr{F}}}}\sqrt{\sum_{j=2}^m\frac{2}{T_n\nu(J_j)}}\leq C\sqrt{\sum_{j=2}^m\frac{2\kappa}{T_n\nu_0(J_j)}}=C\sqrt{\frac{(m-1)2\kappa}{T_n\mu_m}}.$$
- Step 4: Finally, Lemmas \[lemma:ch4wn\] and \[lemma:ch4limitewn\] allow us to conclude that: $$\begin{aligned}
\Delta({\ensuremath {\mathscr{P}}}_n^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0})&=O\bigg(\sqrt{T_n}\sup_{f\in {\ensuremath {\mathscr{F}}}}\big(A_m(f)+B_m(f)+C_m\big)\bigg)\\
& \quad + O\bigg(\sqrt{T_n}\sup_{f\in {\ensuremath {\mathscr{F}}}}\sqrt{\int_{I\setminus{[0,\varepsilon_m]}}\big(f(y)-\hat f_m(y)\big)^2\nu_0(dy)}+\sqrt{\frac{m}{T_n\mu_m}}\bigg).\end{aligned}$$
Proof of Theorem \[ch4teo2\]
----------------------------
Again, before stating some technical lemmas, let us highlight the main ideas of the proof. We recall that the goal is to prove that estimating $f=\frac{d\nu}{d\nu_0}$ from the discrete observations $(X_{t_i})_{i=0}^n$ of a Lévy process without Gaussian component and having Lévy measure $\nu$ is asymptotically equivalent to estimating $f$ from the Gaussian white noise model
$$dy_t=\sqrt{f(t)}dt+\frac{1}{2\sqrt{T_n g(t)}}dW_t,\quad g=\frac{d\nu_0}{d{\ensuremath{\textnormal{Leb}}}},\quad t\in I.$$ Reading ${\ensuremath {\mathscr{P}}}_1 \overset{\Delta} \Longleftrightarrow {\ensuremath {\mathscr{P}}}_2$ as ${\ensuremath {\mathscr{P}}}_1$ is asymptotically equivalent to ${\ensuremath {\mathscr{P}}}_2$, we have:
- Step 1. Clearly $(X_{t_i})_{i=0}^n \overset{\Delta} \Longleftrightarrow (X_{t_i}-X_{t_{i-1}})_{i=1}^n$. Moreover, $(X_{t_i}-X_{t_{i-1}})_i\overset{\Delta} \Longleftrightarrow (\epsilon_iY_i)$ where $(\epsilon_i)$ are i.i.d Bernoulli r.v. with parameter $\alpha=\iota_m \Delta_n e^{-\iota_m\Delta_n}$, $\iota_m:=\int_{I\setminus [0,\varepsilon_m]} f(y)\nu_0(dy)$ and $(Y_i)_i$ are i.i.d. r.v. independent of $(\epsilon_i)_{i=1}^n$ and of density $\frac{ f}{\iota_m}$ with respect to ${\nu_0}_{|_{I\setminus [0,\varepsilon_m]}}$;
- Step 2. $(\epsilon_iY_i)_i \overset{\Delta} \Longleftrightarrow \mathcal M(n;(\gamma_j)_{j=1}^m)$, where $\mathcal M(n;(\gamma_j)_{j=1}^m)$ is a multinomial distribution with $\gamma_1=1-\alpha$ and $\gamma_i:=\alpha\nu(J_i)$ $i=2,\dots,m$;
- Step 3. Gaussian approximation: $\mathcal M(n;(\gamma_1,\dots\gamma_m)) \overset{\Delta} \Longleftrightarrow \bigotimes_{j=2}^m {\ensuremath {\mathscr{Nn}}}(2\sqrt{T_n\nu(J_j)},1)$;
- Step 4. $\bigotimes_{j=2}^m {\ensuremath {\mathscr{Nn}}}(2\sqrt{T_n\nu(J_j)},1)\overset{\Delta} \Longleftrightarrow (y_t)_{t\in I}$.
\[lemma:ch4discreto\] Let $\nu_i$, $i=1,2$, be Lévy measures such that $\nu_1\ll\nu_2$ and $b_1-b_2=\int_{|y|\leq 1}y(\nu_1-\nu_2)(dy)<\infty$. Then, for all $0<t<\infty$, we have: $$\Big\|Q_t^{(b_1,0,\mu_1)}-Q_t^{(b_2,0,\mu_2)}\Big\|_{TV}\leq \sqrt \frac{t}{2} H(\nu_1,\nu_2).$$
For all given $t$, let $K_t$ be the Markov kernel defined as $K_t(\omega,A):={\ensuremath {\mathbb{I}}}_A(\omega_t)$, $\forall \ A\in{\ensuremath {\mathscr{B}}}({\ensuremath {\mathbb{R}}})$, $\forall \ \omega\in D$. Then we have: $$\begin{aligned}
\big\|Q_t^{(b_1,0,\nu_1)}-Q_t^{(b_2,0,\nu_2)}\big\|_{TV}&=\big\|K_tP_t^{(b_1,0,\nu_1)}-K_tP_t^{(b_2,0,\nu_2)}\big\|_{TV}\\
&\leq \big\|P_t^{(b_1,0,\nu_1)}-P_t^{(b_2,0,\nu_2)}\big\|_{TV}\\
&\leq \sqrt \frac{t}{2} H(\nu_1,\nu_2),
\end{aligned}$$ where we have used that Markov kernels reduce the total variation distance and Theorem \[teo:ch4bound\].
\[lemma:ch4bernoulli\] Let $(P_i)_{i=1}^n$, $(Y_i)_{i=1}^n$ and $(\epsilon_i)_{i=1}^n$ be samples of, respectively, Poisson random variables ${\ensuremath {\mathscr{P}}}(\lambda_i)$, random variables with common distribution and Bernoulli random variables of parameters $\lambda_i e^{-\lambda_i}$, which are all independent. Let us denote by $Q_{(Y_i,P_i)}$ (resp. $Q_{(Y_i,\epsilon_i)}$) the law of $\sum_{j=1}^{P_i} Y_j$ (resp., $\epsilon_i Y_i$). Then: $$\label{eq:ch4lambda}
\Big\|\bigotimes_{i=1}^n Q_{(Y_i,P_i)}-\bigotimes_{i=1}^n Q_{(Y_i,\epsilon_i)}\Big\|_{TV}\leq 2\sqrt{\sum_{i=1}^n\lambda_i^2}.$$
The proof of this Lemma can be found in [@esterESAIM], Section 2.1.
\[lemma:ch4troncatura\] Let $f_m^{\textnormal{tr}}$ be the truncated function defined as follows: $$f_m^{\textnormal{tr}}(x)=\begin{cases}
1 &\mbox{ if } x\in[0,\varepsilon_m]\\
f(x) &\mbox{ otherwise}
\end{cases}$$ and let $\nu_m^{\textnormal{tr}}$ (resp. $\nu_m^{\textnormal{res}}$) be the Lévy measure having $f_m^{\textnormal{tr}}$ (resp. ${f|_{I\setminus [0,\varepsilon_m]}}$) as a density with respect to $\nu_0$. Denote by ${\ensuremath {\mathscr{Q}}}_{n}^{\textnormal{tr},\nu_0}$ the statistical model associated with the family of probabilities $\Big(\bigotimes_{i=1}^nQ_{t_i-t_{i-1}}^{(\gamma^{\nu_m^{\textnormal{tr}}-\nu_0},0,\nu_m^{\textnormal{tr}})}:\frac{d\nu_m^{\textnormal{tr}}}{d\nu_0}\in{\ensuremath {\mathscr{F}}}\Big)$ and by ${\ensuremath {\mathscr{Q}}}_{n}^{\textnormal{res},\nu_0}$ the model associated with the family of probabilities $\Big(\bigotimes_{i=1}^nQ_{t_i-t_{i-1}}^{(\gamma^{\nu_m^{\textnormal{res}}-\nu_0},0,\nu_m^{\textnormal{res}})}:\frac{d\nu_m^{\textnormal{res}}}{d\nu_0}\in{\ensuremath {\mathscr{F}}}\Big)$. Then: $$\Delta({\ensuremath {\mathscr{Q}}}_{n}^{\textnormal{tr},\nu_0},{\ensuremath {\mathscr{Q}}}_{n}^{\textnormal{res},\nu_0})=0.$$
Let us start by proving that $\delta({\ensuremath {\mathscr{Q}}}_{n}^{\textnormal{tr},\nu_0},{\ensuremath {\mathscr{Q}}}_{n}^{\textnormal{res},\nu_0})=0.$ For that, let us consider two independent Lévy processes, $X^{\textnormal{tr}}$ and $X^0$, of Lévy triplets given by $\big(\gamma^{\nu_m^{\textnormal{tr}}-\nu_0},0,\nu_m^{\textnormal{tr}-\nu_0}\big)$ and $\big(0,0,\nu_0|_{[0,\varepsilon_m]}\big)$, respectively. Then it is clear (using the *Lévy-Khintchine formula*) that the random variable $X_t^{\textnormal{tr}}- X_t^0$ is a randomization of $X_t^{\textnormal{tr}}$ (since the law of $X_t^0$ does not depend on $\nu$) having law $Q_t^{(\gamma^{\nu_m^{\textnormal{res}}-\nu_0},0,\nu_m^{\textnormal{res}})}$, for all $t\geq 0$. Similarly, one can prove that $\delta({\ensuremath {\mathscr{Q}}}_{n}^{\textnormal{res},\nu_0},{\ensuremath {\mathscr{Q}}}_{n}^{\textnormal{tr},\nu_0})=0.$
As a preliminary remark, observe that the model ${\ensuremath {\mathscr{Q}}}_n^{\nu_0}$ is equivalent to the one that observes the increments of $\big((x_t),P_{T_n}^{(\gamma^{\nu-\nu_0},0,\nu)}\big)$, that is, the model $\tilde{\ensuremath {\mathscr{Q}}}_n^{\nu_0}$ associated with the family of probabilities $\Big(\bigotimes_{i=1}^nQ_{t_i-t_{i-1}}^{(\gamma^{\nu-\nu_0},0,\nu)}:\frac{d\nu}{d\nu_0}\in{\ensuremath {\mathscr{F}}}\Big)$.
- Step 1: Facts \[ch4h\]–\[ch4hp\] and Lemma \[lemma:ch4discreto\] allow us to write $$\begin{aligned}
&\Big\|\bigotimes_{i=1}^nQ_{\Delta_n}^{(\gamma^{\nu-\nu_0},0,\nu)}-\bigotimes_{i=1}^nQ_{\Delta_n}^{(\gamma^{\nu_m^{\textnormal{tr}}-\nu_0},0, \nu_m^{\textnormal{tr}})}\Big\|_{TV}\leq \sqrt{n\sqrt\frac{\Delta_n}{2}H(\nu,\nu_m^{\textnormal{tr}})}\\&=\sqrt{n\sqrt\frac{\Delta_n}{2}\sqrt{\int_0^{\varepsilon_m}\big(\sqrt{f(y)}-1\big)^2\nu_0(dy)}}.\end{aligned}$$ Using this bound together with Lemma \[lemma:ch4troncatura\] and the notation therein, we get $\Delta({\ensuremath {\mathscr{Q}}}_n^{\nu_0}, {\ensuremath {\mathscr{Q}}}_{n}^{\textnormal{res},\nu_0})\leq \sqrt{n\sqrt\frac{\Delta_n}{2}\sup_{f\in {\ensuremath {\mathscr{F}}}}H(f, f_m^{\textnormal{tr}})}$. Observe that $\nu_m^{\textnormal{res}}$ is a finite Lévy measure, hence $\Big((x_t),P_{T_n}^{(\gamma^{\nu_m^{\textnormal{res}}},0,\nu_m^{\textnormal{res}})}\Big)$ is a compound Poisson process with intensity equal to $\iota_m:=\int_{I\setminus [0,\varepsilon_m]} f(y)\nu_0(dy)$ and jumps size density $\frac{ f(x)g(x)}{\iota_m}$, for all $x\in I\setminus [0,\varepsilon_m]$ (recall that we are assuming that $\nu_0$ has a density $g$ with respect to Lebesgue). In particular, this means that $Q_{\Delta_n}^{(\gamma^{\nu_m^{\textnormal{res}}},0,\nu_m^{\textnormal{res}})}$ can be seen as the law of the random variable $\sum_{j=1}^{P_i}Y_j$ where $P_i$ is a Poisson variable of mean $\iota_m \Delta_n$, independent from $(Y_i)_{i\geq 0}$, a sequence of i.i.d. random variables with density $\frac{ fg}{\iota_m}{\ensuremath {\mathbb{I}}}_{I\setminus[0,\varepsilon_m]}$ with respect to Lebesgue. Remark also that $\iota_m$ is confined between $\kappa \nu_0\big(I\setminus [0,\varepsilon_m]\big)$ and $M\nu_0\big(I\setminus [0,\varepsilon_m]
\big)$.
Let $(\epsilon_i)_{i\geq 0}$ be a sequence of i.i.d. Bernoulli variables, independent of $(Y_i)_{i\geq 0}$, with mean $\iota_m \Delta_n e^{-\iota_m\Delta_n}$. For $i=1,\dots,n$, denote by $Q_i^{\epsilon,f}$ the law of the variable $\epsilon_iY_i$ and by ${\ensuremath {\mathscr{Q}}}_n^{\epsilon}$ the statistical model associated with the observations of the vector $(\epsilon_1Y_1,\dots,\epsilon_nY_n)$, i.e. $${\ensuremath {\mathscr{Q}}}_n^{\epsilon}=\bigg(I^n,{\ensuremath {\mathscr{B}}}(I^n),\bigg\{\bigotimes_{i=1}^n Q_i^{\epsilon,f}:f\in{\ensuremath {\mathscr{F}}}\bigg\}\bigg).$$ Furthermore, denote by $\tilde Q_i^f$ the law of $\sum_{j=1}^{P_i}Y_j$. Then an application of Lemma \[lemma:ch4bernoulli\] yields: $$\begin{aligned}
\Big\|\bigotimes_{i=1}^n\tilde Q_i^f&-\bigotimes_{i=1}^nQ_i^{\epsilon,f}\Big\|_{TV} \leq 2\iota_m\sqrt{n\Delta_n^2}\leq 2M\nu_0\big(I\setminus [0,\varepsilon_m]\big)\sqrt{n\Delta_n^2}.\end{aligned}$$ Hence, we get: $$\label{eq:ch4bernoulli}
\Delta({\ensuremath {\mathscr{Q}}}_{n}^{\textnormal{res},\nu_0},{\ensuremath {\mathscr{Q}}}_n^{\epsilon})=O\bigg(\nu_0\big(I\setminus [0,\varepsilon_m]\big)\sqrt{n\Delta_n^2}\bigg).$$ Here the O depends only on $M$.
- Step 2: Let us introduce the following random variables: $$Z_1=\sum_{j=1}^n{\ensuremath {\mathbb{I}}}_{\{0\}}(\epsilon_jY_j); \quad Z_i=\sum_{j=1}^n{\ensuremath {\mathbb{I}}}_{J_i}(\epsilon_jY_j),\ i=2,\dots,m.$$ Observe that the law of the vector $(Z_1,\dots,Z_m)$ is multinomial $\mathcal M(n;\gamma_1,\dots,\gamma_m)$ where $$\gamma_1=1-\iota_m \Delta_n e^{-\iota_m \Delta_n},\quad \gamma_i=\Delta_n e^{-\iota_m \Delta_n}\nu(J_i),\quad i=2,\dots,m.$$ Let us denote by $\mathcal M_n$ the statistical model associated with the observation of $(Z_1,\dots,Z_m)$. Clearly $\delta({\ensuremath {\mathscr{Q}}}_n^{\epsilon},\mathcal M_n)=0$. Indeed, $\mathcal M_n$ is the image experiment by the random variable $S:I^n\to\{1,\dots,n\}^{m}$ defined as $$S(x_1,\dots,x_n)=\Big(\#\{j: x_j=0\}; \#\big\{j: x_j\in J_2\big\};\dots;\#\big\{j: x_j\in J_m\big\}\Big),$$ where $\# A$ denotes the cardinal of the set $A$.
We shall now prove that $\delta(\mathcal M_n,{\ensuremath {\mathscr{Q}}}_n^{\epsilon}) \leq \sup_{f\in{\ensuremath {\mathscr{F}}}}\sqrt{n\Delta_n H^2(f,\hat f_m)}$. We start by defining a discrete random variable $X^*$ concentrated at the points $0$, $x_i^*$, $i=2,\dots,m$: $${\ensuremath {\mathbb{P}}}(X^*=y)=\begin{cases}
\gamma_i &\mbox{ if } y=x_i^*,\quad i=1,\dots,m,\\
0 &\mbox{ otherwise},
\end{cases}$$ with the convention $x_1^*=0$. It is easy to see that $\mathcal M_n$ is equivalent to the statistical model associated with $n$ independent copies of $X^*$. Let us introduce the Markov kernel $$K(x_i^*, A) = \begin{cases}
{\ensuremath {\mathbb{I}}}_A(0) & \text{if } i = 1,\\
\int_A V_i(x) \nu_0(dx) & \text{otherwise.}
\end{cases}$$ Denote by $P^*$ the law of the random variable $X^*$ and by $Q_i^{\epsilon,\hat f}$ the law of a random variable $\epsilon_i \hat Y_i$ where $\epsilon_i$ is Bernoulli independent of $\hat Y_i$, with mean $\iota_m\Delta_n e^{-\iota_m\Delta_n}$ and $\hat Y_i$ has a density $\frac{\hat f_m g}{\iota_m}{\ensuremath {\mathbb{I}}}_{I\setminus[0,\varepsilon_m]}$ with respect to Lebesgue. The same computations as in Lemma \[lemma:ch4kernel\] prove that $KP^*=Q_i^{\epsilon,\hat f}$. Hence, thanks to Remark \[ch4independentkernels\], we get the equivalence between $\mathcal M_n$ and the statistical model associated with the observations of $n$ independent copies of $\epsilon_i \hat Y_i$. In order to bound $\delta(\mathcal M_n,{\ensuremath {\mathscr{Q}}}_n^{\epsilon})$ it is enough to bound the total variation distance between the probabilities $\bigotimes_{i=1}^n Q_i^{\epsilon,f}$ and $\bigotimes_{i=1}^n Q_i^{\epsilon,\hat f}$. Alternatively, we can bound the Hellinger distance between each of the $Q_i^{\epsilon,f}$ and $Q_i^{\epsilon,\hat f}$, thanks to Facts \[ch4h\] and \[ch4hp\], which is: $$\begin{aligned}
\bigg\|\bigotimes_{i=1}^nQ_i^{\epsilon,f} -\bigotimes_{i=1}^nQ_i^{\epsilon,\hat f}\bigg\|_{TV} &\leq \sqrt{\sum_{i=1}^n H^2\big(Q_i^{\epsilon,f}, Q_i^{\epsilon,\hat f}\big)}\\
&= \sqrt{\sum_{i=1}^n \frac{1-\gamma_1}{\iota} H^2(f, \hat f_m)} \leq \sqrt{n\Delta_n H^2(f, \hat f_m)}.\end{aligned}$$ It follows that $$\delta(\mathcal M_n,{\ensuremath {\mathscr{Q}}}_n^{\epsilon})\leq \sqrt{n\Delta_n} \sup_{f \in {\ensuremath {\mathscr{F}}}}H(f,\hat f_m).$$
- Step 3: Let us denote by $\mathcal N_m^*$ the statistical model associated with the observation of $m$ independent Gaussian variables ${\ensuremath {\mathscr{Nn}}}(n\gamma_i,n\gamma_i)$, $i=1,\dots,m$. Very similar computations to those in [@cmultinomial] yield $$\Delta(\mathcal M_n,\mathcal N_m^*)=O\Big(\frac{m \ln m}{\sqrt{n}}\Big).$$ In order to prove the asymptotic equivalence between $\mathcal M_n$ and $\mathcal N_m$ defined as in we need to introduce some auxiliary statistical models. Let us denote by $\mathcal A_m$ the experiment obtained from $\mathcal{N}_m^*$ by disregarding the first component and by $\mathcal V_m$ the statistical model associated with the multivariate normal distribution with the same means and covariances as a multinomial distribution $\mathcal M(n,\gamma_1,\dots,\gamma_m)$. Furthermore, let us denote by $\mathcal N_m^{\#}$ the experiment associated with the observation of $m-1$ independent Gaussian variables ${\ensuremath {\mathscr{Nn}}}(\sqrt{n\gamma_i},\frac{1}{4})$, $i=2,\dots,m$. Clearly $\Delta(\mathcal V_m,\mathcal A_m)=0$ for all $m$: In one direction one only has to consider the projection disregarding the first component; in the other direction, it is enough to remark that $\mathcal V_m$ is the image experiment of $\mathcal A_m$ by the random variable $S:(x_2,\dots,x_m)\to (n(1-\frac{\sum_{i=2}^m x_i}{n}),x_2,\dots,x_m)$. Moreover, using two results contained in [@cmultinomial], see Sections 7.1 and 7.2, one has that $$\Delta(\mathcal A_m,\mathcal N_m^*)=O\bigg(\sqrt{\frac{m}{n}}\bigg),\quad \Delta(\mathcal A_m,\mathcal N_m^{\#})=O\bigg(\frac{m}{\sqrt n}\bigg).$$ Finally, using Facts \[ch4h\] and \[fact:ch4gaussiane\] we can write $$\begin{aligned}
\Delta(\mathcal N_m^{\#},\mathcal N_m)&\leq \sqrt{2\sum_{i=2}^m \Big(\sqrt{T_n\nu(J_i)}-\sqrt{T_n\nu(J_i)\exp(-\iota_m\Delta_n)}\Big)^2}\\
&\leq\sqrt{2T_n\Delta_n^2\iota_m^3}\leq \sqrt{2n\Delta_n^3M^3\big(\nu_0\big(I\setminus [0,\varepsilon_m]\big)\big)^3}.
\end{aligned}$$ To sum up, $\Delta(\mathcal M_n,\mathcal N_m)=O\Big(\frac{m \ln m}{\sqrt{n}}+\sqrt{n\Delta_n^3\big(\nu_0\big(I\setminus [0,\varepsilon_m]\big)\big)^3}\Big)$, with the $O$ depending only on $\kappa$ and $M$.
- Step 4: An application of Lemmas \[lemma:ch4wn\] and \[lemma:ch4limitewn\] yield $$\Delta(\mathcal N_m,{\ensuremath {\mathscr{W}}}_n^{\nu_0}) \leq 2\sqrt T_n \sup_{f\in{\ensuremath {\mathscr{F}}}} \big(A_m(f)+B_m(f)+C_m(f)\big).$$
Proofs of the examples
======================
The purpose of this section is to give detailed proofs of Examples \[ex:ch4esempi\] and Examples \[ex:ch4CPP\]–\[ex3\]. As in Section \[sec:ch4proofs\] we suppose $I\subseteq {\ensuremath {\mathbb{R}}}_+$. We start by giving some bounds for the quantities $A_m(f)$, $B_m(f)$ and $L_2(f, \hat f_m)$, the $L_2$-distance between the restriction of $f$ and $\hat f_m$ on $I\setminus[0,\varepsilon_m].$
Bounds for $A_m(f)$, $B_m(f)$, $L_2(f, \hat{f}_m)$ when $\hat f_m$ is piecewise linear.
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In this section we suppose $f$ to be in ${\ensuremath {\mathscr{F}}}_{(\gamma, K, \kappa, M)}^I$ defined as in . We are going to assume that the $V_j$ are given by triangular/trapezoidal functions as in . In particular, in this case $\hat f_m$ is piecewise linear.
\[lemma:ch4hellinger\] Let $0<\kappa < M$ be two constants and let $f_i$, $i=1,2$ be functions defined on an interval $J$ and such that $\kappa \leq f_i\leq M$, $i=1,2$. Then, for any measure $\nu_0$, we have: $$\begin{aligned}
\frac{1}{4 M} \int_J \big(f_1(x)-f_2(x)\big)^2 \nu_0(dx)&\leq\int_J \big(\sqrt{f_1(x)} - \sqrt{f_2(x)}\big)^2\nu_0(dx)\\
&\leq \frac{1}{4 \kappa} \int_J \big(f_1(x)-f_2(x)\big)^2\nu_0(dx).
\end{aligned}$$
This simply comes from the following inequalities: $$\begin{aligned}
\frac{1}{2\sqrt M} (f_1(x)-f_2(x)) &\leq \frac{f_1(x)-f_2(x)}{\sqrt{f_1(x)}+\sqrt{f_2(x)}} = \sqrt{f_1(x)} - \sqrt{f_2(x)}\\
&\leq \frac{1}{2 \sqrt{\kappa}} (f_1(x)-f_2(x)).
\end{aligned}$$
Recall that $x_i^*$ is chosen so that $\int_{J_i} (x-x_i^*) \nu_0(dx) = 0$. Consider the following Taylor expansions for $x \in J_i$: $$f(x) = f(x_i^*) + f'(x_i^*) (x-x_i^*) + R_i(x); \quad \hat{f}_m(x) = \hat{f}_m(x_i^*) + \hat{f}_m'(x_i^*) (x-x_i^*),$$ where $\hat{f}_m(x_i^*) = \frac{\nu(J_i)}{\nu_0(J_i)}$ and $\hat{f}_m'(x_i^*)$ is the left or right derivative in $x_i^*$ depending whether $x < x_i^*$ or $x > x_i^*$ (as $\hat f_m$ is piecewise linear, no rest is involved in its Taylor expansion).
\[lemma:ch4bounds\] The following estimates hold: $$\begin{aligned}
|R_i(x)| &\leq K |\xi_i - x_i^*|^\gamma |x-x_i^*|; \\
\big|f(x_i^*) - \hat{f}_m(x_i^*)\big| &\leq \|R_i\|_{L_\infty(\nu_0)} \text{ for } i = 2, \dots, m-1; \label{eqn:bounds}\\
\big|f(x)-\hat{f}_m(x)\big| &\leq
\begin{cases}
2 \|R_i\|_{L_\infty(\nu_0)} + K |x_i^*-\eta_i|^\gamma |x-x_i^*| & \text{ if } x \in J_i, \ i = 3, \dots, m-1;\\
C |x-\tau_i| & \text { if } x \in J_i, \ i \in \{2, m\}.
\end{cases}
\end{aligned}$$ for some constant $C$ and points $\xi_i \in J_i$, $\eta_i\in J_{i-1} \cup J_i\cup J_{i+1}$, $\tau_2 \in J_2 \cup J_3$ and $\tau_m \in J_{m-1} \cup J_m$.
By definition of $R_i$, we have $$|R_i(x)| = \Big| \big(f'(\xi_i) - f'(x_i^*)\big)(x-x_i^*) \Big| \leq K |\xi_i - x_i^*|^\gamma |x-x_i^*|,$$ for some point $\xi_i \in J_i$. For the second inequality, $$\begin{aligned}
|f(x_i^*)-\hat{f}_m(x_i^*)| &= \frac{1}{\nu_0(J_i)} \Big| \int_{J_i} (f(x_i^*)-f(x)) \nu_0(dx)\Big|\\
&= \frac{1}{\nu_0(J_i)} \bigg|\int_{J_i} R_i(x) \nu_0(dx)\bigg| \leq \|R_i\|_{L_\infty(\nu_0)},
\end{aligned}$$ where in the first inequality we have used the defining property of $x_i^*$. For the third inequality, let us start by proving that for all $2 < i < m-1$, $\hat{f}_m'(x_i^*) = f'(\chi_i)$ for some $\chi_i \in J_i\cup J_{i+1}$ (here, we are considering right derivatives; for left ones, this would be $J_{i-1} \cup J_i$). To see that, take $x\in J_i\cap [x_i^*,x_{i+1}^*]$ and introduce the function $h(x):=f(x)-l(x)$ where $$l(x)=\frac{x-x_i^*}{x_{i+1}^*-x_i^*}\big(\hat f_m(x_{i+1}^*)-\hat f_m(x_i^*)\big)+\hat f_m(x_i^*).$$ Then, using the fact that $\int_{J_i}(x-x_i^*)\nu_0(dx)=0$ joint with $\int_{J_{i+1}}(x-x_{i+1}^*)\nu_0(dx)=(x_{j+1}^*-x_j^*)\mu_m$, we get $$\int_{J_i}h(x)\nu_0(dx)=0=\int_{J_{i+1}}h(x)\nu_0(dx).$$ In particular, by means of the mean theorem, one can conclude that there exist two points $p_i\in J_i$ and $p_{i+1}\in J_{i+1}$ such that $$h(p_i)=\frac{\int_{J_i}h(x)\nu_0(dx)}{\nu_0(J_i)}=\frac{\int_{J_{i+1}}h(x)\nu_0(dx)}{\nu_0(J_{i+1})}=h(p_{i+1}).$$ As a consequence, we can deduce that there exists $\chi_i\in[p_i,p_{i+1}]\subseteq J_i\cup J_{i+1}$ such that $h'(\chi_i)=0$, hence $f'(\chi_i)=l'(\chi_i)=\hat f_m'(x_i^*)$. When $2 < i < m-1$, the two Taylor expansions joint with the fact that $\hat{f}_m'(x_i^*) = f'(\chi_i)$ for some $\chi_i \in J_i\cup J_{i+1}$, give $$\begin{aligned}
|f(x) - \hat{f}_m (x)| &\leq |f(x_i^*) - \hat{f}_m(x_i^*)| + |R_i(x)| + K |x_i^* - \chi_i|^\gamma |x-x_i^*|\\
& \leq 2 \|R_i\|_{L_\infty(\nu_0)} + K |x_i^* - \chi_i|^\gamma |x-x_i^*|
\end{aligned}$$ whenever $x \in J_i$ and $x > x_i^*$ (the case $x < x_i^*$ is handled similarly using the left derivative of $\hat f_m$ and $\xi_i \in J_{i-1} \cup J_i$). For the remaining cases, consider for example $i = 2$. Then $\hat{f}_m(x)$ is bounded by the minimum and the maximum of $f$ on $J_2 \cup J_3$, hence $\hat{f}_m(x) = f(\tau)$ for some $\tau \in J_2 \cup J_3$. Since $f'$ is bounded by $C = 2M +K$, one has $|f(x) - \hat{f}_m(x)| \leq C|x-\tau|$.
\[lemma:ch4abc\] With the same notations as in Lemma \[lemma:ch4bounds\], the estimates for $A_m^2(f)$, $B_m^2(f)$ and $L_2(f, \hat{f}_m)^2$ are as follows: $$\begin{aligned}
L_2(f, \hat{f}_m)^2&\leq \frac{1}{4\kappa} \bigg( \sum_{i=3}^m \int_{J_i} \Big(2 \|R_i\|_{L_\infty(\nu_0)} + K |x_i^*-\eta_i|^\gamma|x-x_i^*|\Big)^2 \nu_0(dx) \\
&\phantom{=}\ + C^2 \Big(\int_{J_2}|x-\tau_2|^2\nu_0(dx) + \int_{J_m}|x-\tau_m|^2\nu_0(dx)\Big).\\
A_m^2(f) &= L_2\big(\sqrt{f}, \widehat{\sqrt{f}}_m\big)^2 = O\Big(L_2(f, \hat{f}_m)^2\Big)\\
B_m^2(f) &= O\bigg( \sum_{i=2}^{m} \frac{1}{\sqrt{\kappa}} \nu_0(J_i) (2 \sqrt{M} + 1)^2 \|R_i\|_{L_\infty(\nu_0)}^2\bigg).
\end{aligned}$$
The $L_2$-bound is now a straightforward application of Lemmas \[lemma:ch4hellinger\] and \[lemma:ch4bounds\]. The one on $A_m(f)$ follows, since if $f \in {\ensuremath {\mathscr{F}}}_{(\gamma, K, \kappa, M)}^I$ then $\sqrt{f} \in {\ensuremath {\mathscr{F}}}_{(\gamma, \frac{K}{\sqrt{\kappa}}, \sqrt{\kappa}, \sqrt{M})}^I$. In order to bound $B_m^2(f)$ write it as: $$B_m^2(f)=\sum_{j=1}^m \nu_0(J_j)\bigg(\frac{\int_{J_j}\sqrt{f(y)}\nu_0(dy)}{\nu_0(J_j)}-\sqrt{\frac{\nu(J_j)}{\nu_0(J_j)}}\bigg)^2=:\sum_{j=1}^m \nu_0(J_j)E_j^2.$$ By the triangular inequality, let us bound $E_j$ by $F_j+G_j$ where: $$F_j=\bigg|\sqrt{\frac{\nu(J_j)}{\nu_0(J_j)}}-\sqrt{f(x_j^*)}\bigg| \quad \textnormal{ and }\quad G_j=\bigg|\sqrt{f(x_j^*)}-\frac{\int_{J_j}\sqrt{f(y)}\nu_0(dy)}{\nu_0(J_j)}\bigg|.$$ Using the same trick as in the proof of Lemma \[lemma:ch4hellinger\], we can bound: $$\begin{aligned}
F_j \leq 2 \sqrt{M} \bigg|\frac{\int_{J_j} \big(f(x)-f(x_i^*)\big)\nu_0(dx)}{\nu_0(J_j)}\bigg| \leq 2 \sqrt{M} \|R_j\|_{L_\infty(\nu_0)}.
\end{aligned}$$ On the other hand, $$\begin{aligned}
G_j&=\frac{1}{\nu_0(J_j)}\bigg|\int_{J_j}\big(\sqrt{f(x_j^*)}-\sqrt{f(y)}\big)\nu_0(dy)\bigg|\\
&=\frac{1}{\nu_0(J_j)}\bigg|\int_{J_j}\bigg(\frac{f'(x_j^*)}{2\sqrt{f(x_j^*)}}(x-x_j^*)+\tilde R_j(y)\bigg)\nu_0(dy)\bigg| \leq \|\tilde R_j\|_{L_\infty(\nu_0)},
\end{aligned}$$ which has the same magnitude as $\frac{1}{\kappa}\|R_j\|_{L_\infty(\nu_0)}$.
Observe that when $\nu_0$ is finite, there is no need for a special definition of $\hat{f}_m$ near $0$, and all the estimates in Lemma \[lemma:ch4bounds\] hold true replacing every occurrence of $i = 2$ by $i = 1$.
\[rmk:nonlinear\] The same computations as in Lemmas \[lemma:ch4bounds\] and \[lemma:ch4abc\] can be adapted to the general case where the $V_j$’s (and hence $\hat f_m$) are not piecewise linear. In the general case, the Taylor expansion of $\hat f_m$ in $x_i^*$ involves a rest as well, say $\hat R_i$, and one needs to bound this, as well.
Proofs of Examples \[ex:ch4esempi\] {#subsec:esempi}
-----------------------------------
In the following, we collect the details of the proofs of Examples \[ex:ch4esempi\].
**1. The finite case:** $\nu_0\equiv {\ensuremath{\textnormal{Leb}}}([0,1])$.
Remark that in the case where $\nu_0$ if finite there are no convergence problems near zero and so we can consider the easier approximation of $f$: $$\hat f_m(x):=
\begin{cases}
m\theta_1 & \textnormal{if } x\in \big[0,x_1^*\big],\\
m^2\big[\theta_{j+1}(x-x_j^*)+\theta_j(x_{j+1}^*-x)\big] & \textnormal{if } x\in (x_j^*,x_{j+1}^*] \quad j = 1,\dots,m-1,\\
m\theta_m & \textnormal{if } x\in (x_m^*,1]
\end{cases}$$ where $$x_j^*=\frac{2j-1}{2m},\quad J_j=\Big(\frac{j-1}{m},\frac{j}{m}\Big],\quad \theta_j=\int_{J_j}f(x)dx, \quad j=1,\dots,m.$$ In this case we take $\varepsilon_m = 0$ and Conditions $(C2)$ and $(C2')$ coincide: $$\lim_{n\to\infty}n\Delta_n\sup_{f\in {\ensuremath {\mathscr{F}}}}\Big(A_m^2(f)+B_m^2(f)\Big) = 0.$$ Applying Lemma \[lemma:ch4abc\], we get $$\sup_{f\in {\ensuremath {\mathscr{F}}}} \Big(L_2(f,\hat f_m)+ A_m(f)+ B_m(f)\Big)= O\big(m^{-\frac{3}{2}}+m^{-1-\gamma}\big);$$ (actually, each of the three terms on the left hand side has the same rate of convergence).
**2. The finite variation case:** $\frac{d\nu_0}{d{\ensuremath{\textnormal{Leb}}}}(x)=x^{-1}{\ensuremath {\mathbb{I}}}_{[0,1]}(x).$
To prove that the standard choice of $V_j$ described at the beginning of Examples \[ex:ch4esempi\] leads to $\displaystyle{\int_{\varepsilon_m}^1 V_j(x)\frac{dx}{x}=1}$, it is enough to prove that this integral is independent of $j$, since in general $\displaystyle{\int_{\varepsilon_m}^1 \sum_{j=2}^m V_j(x)\frac{dx}{x}=m-1}.$ To that aim observe that, for $j=3,\dots,m-1$, $$\mu_m\int_{\varepsilon_m}^1 V_j(x)\nu_0(dx)=\int_{x_{j-1}^*}^{x_j^*}\frac{x-x_{j-1}^*}{x_j^*-x_{j-1}^*}\frac{dx}{x}+\int_{x_j^*}^{x_{j+1}^*}\frac{x_{j+1}^*-x}{x_{j+1}^*-x_j^*}\frac{dx}{x}.$$ Let us show that the first addendum does not depend on $j$. We have $$\int_{x_{j-1}^*}^{x_j^*}\frac{dx}{x_j^*-x_{j-1}^*}=1\quad \textnormal{and}\quad -\frac{x_{j-1}^*}{x_j^*-x_{j-1}^*}\int_{x_{j-1}^*}^{x_j^*}\frac{dx}{x}=\frac{x_{j-1}^*}{x_j^*-x_{j-1}^*}\ln\Big(\frac{x_{j-1}^*}{x_j^*}\Big).$$ Since $x_j^*=\frac{v_j-v_{j-1}}{\mu_m}$ and $v_j=\varepsilon_m^{\frac{m-j}{m-1}}$, the quantities $\frac{x_j^*}{x_{j-1}^*}$ and, hence, $\frac{x_{j-1}^*}{x_j^*-x_{j-1}^*}$ do not depend on $j$. The second addendum and the trapezoidal functions $V_2$ and $V_m$ are handled similarly. Thus, $\hat f_m$ can be chosen of the form
$$\hat f_m(x):=
\begin{cases}
\quad 1 & \textnormal{if } x\in \big[0,\varepsilon_m\big],\\
\frac{\nu(J_2)}{\mu_m} & \textnormal{if } x\in \big(\varepsilon_m, x_2^*\big],\\
\frac{1}{x_{j+1}^*-x_j^*}\bigg[\frac{\nu(J_{j+1})}{\mu_m}(x-x_j^*)+\frac{\nu(J_{j})}{\mu_m}(x_{j+1}^*-x)\bigg] & \textnormal{if } x\in (x_j^*,x_{j+1}^*] \quad j = 2,\dots,m-1,\\
\frac{\nu(J_m)}{\mu_m} & \textnormal{if } x\in (x_m^*,1].
\end{cases}$$
A straightforward application of Lemmas \[lemma:ch4bounds\] and \[lemma:ch4abc\] gives $$\sqrt{\int_{\varepsilon_m}^1\Big(f(x)-\hat f_m(x)\Big)^2 \nu_0(dx)} +A_m(f)+B_m(f)=O\bigg(\bigg(\frac{\ln m}{m}\bigg)^{\gamma+1} \sqrt{\ln (\varepsilon_m^{-1})}\bigg),$$ as announced.
**3. The infinite variation, non-compactly supported case:** $\frac{d\nu_0}{d{\ensuremath{\textnormal{Leb}}}}(x)=x^{-2}{\ensuremath {\mathbb{I}}}_{{\ensuremath {\mathbb{R}}}_+}(x)$. Recall that we want to prove that $$L_2(f,\hat f_m)^2+A_m^2(f)+B_m^2(f)=O\bigg(\frac{H(m)^{3+4\gamma}}{(\varepsilon_m m)^{2\gamma}}+\sup_{x\geq H(m)}\frac{f(x)^2}{H(m)}\bigg),$$ for any given sequence $H(m)$ going to infinity as $m\to\infty$.
Let us start by addressing the problem that the triangular/trapezoidal choice for $V_j$ is not doable. Introduce the following notation: $V_j = {\ensuremath {\accentset{\triangle}{V}}}_j + A_j$, $j = 2, \dots, m$, where the ${\ensuremath {\accentset{\triangle}{V}}}_j$’s are triangular/trapezoidal function similar to those in . The difference is that here, since $x_m^*$ is not defined, ${\ensuremath {\accentset{\triangle}{V}}}_{m-1}$ is a trapezoid, linear between $x_{m-2}^*$ and $x_{m-1}^*$ and constantly equal to $\frac{1}{\mu_m}$ on $[x_{m-1}^*,v_{m-1}]$ and ${\ensuremath {\accentset{\triangle}{V}}}_m$ is supported on $[v_{m-1},\infty)$, where it is constantly equal to $\frac{1}{\mu_m}$. Each $A_j$ is chosen so that:
1. It is supported on $[x_{j-1}^*, x_{j+1}^*]$ (unless $j = 2$, $j = m-1$ or $j = m$; in the first case the support is $[x_2^*, x_3^*]$, in the second one it is $[x_{m-2}^*, x_{m-1}^*]$, and $A_m \equiv 0$);
2. ${A_j}$ coincides with $-A_{j-1}$ on $[x_{j-1}^*, x_j^*]$, $j = 3, \dots, m-1$ (so that $\sum V_j \equiv \frac{1}{\mu_n}$) and its first derivative is bounded (in absolute value) by $\frac{1}{\mu_m(x_j^* - x_{j-1}^*)}$ (so that $V_j$ is non-negative and bounded by $\frac{1}{\mu_n}$);
3. $A_j$ vanishes, along with its first derivatives, on $x_{j-1}^*$, $x_j^*$ and $x_{j+1}^*$.
We claim that these conditions are sufficient to assure that $\hat f_m$ converges to $f$ quickly enough. First of all, by Remark \[rmk:nonlinear\], we observe that, to have a good bound on $L_2(f, \hat f_m)$, the crucial property of $\hat f_m$ is that its first right (resp. left) derivative has to be equal to $\frac{1}{\mu_m(x_{j+1}^*-x_j^*)}$ (resp. $\frac{1}{\mu_m(x_{j}^*-x_{j-1}^*)}$) and its second derivative has to be small enough (for example, so that the rest $\hat R_j$ is as small as the rest $R_j$ of $f$ already appearing in Lemma \[lemma:ch4bounds\]).
The (say) left derivatives in $x_j^*$ of $\hat f_m$ are given by $$\hat f_m'(x_j^*) = \big({\ensuremath {\accentset{\triangle}{V}}}_j'(x_j^*) + A_j'(x_j^*)\big) \big(\nu(J_j)-\nu(J_{j-1})\big); \quad \hat f_m''(x_j^*) = A_j''(x_j^*)\big(\nu(J_j)-\nu(J_{j-1})\big).$$ Then, in order to bound $|\hat f_m''(x_j^*)|$ it is enough to bound $|A_j''(x_j^*)|$ because: $$\big|\hat f_m''(x_j^*)\big| \leq |A_j''(x_j^*)| \Big|\int_{J_j} f(x) \frac{dx}{x^2} - \int_{J_{j-1}} f(x) \frac{dx}{x^2}\Big| \leq |A_j''(x_j^*)| \displaystyle{\sup_{x\in I}}|f'(x)|(\ell_{j}+\ell_{j-1}) \mu_m,$$ where $\ell_{j}$ is the Lebesgue measure of $J_{j}$.
We are thus left to show that we can choose the $A_j$’s satisfying points 1-3, with a small enough second derivative, and such that $\int_I V_j(x) \frac{dx}{x^2} = 1$. To make computations easier, we will make the following explicit choice: $$A_j(x) = b_j (x-x_j^*)^2 (x-x_{j-1}^*)^2 \quad \forall x \in [x_{j-1}^*, x_j^*),$$ for some $b_j$ depending only on $j$ and $m$ (the definitions on $[x_j^*, x_{j+1}^*)$ are uniquely determined by the condition $A_j + A_{j+1} \equiv 0$ there).
Define $j_{\max}$ as the index such that $H(m) \in J_{j_{\max}}$; it is straightforward to check that $$j_{\max} \sim m- \frac{\varepsilon_m(m-1)}{H(m)}; \quad x_{m-k}^* = \varepsilon_m(m-1) \log \Big(1+\frac{1}{k}\Big), \quad k = 1, \dots, m-2.$$ One may compute the following Taylor expansions: $$\begin{aligned}
\int_{x_{m-k-1}^*}^{x_{m-k}^*} {\ensuremath {\accentset{\triangle}{V}}}_{m-k}(x) \nu_0(dx) &= \frac{1}{2} - \frac{1}{6k} + \frac{5}{24k^2} + O\Big(\frac{1}{k^3}\Big);\\
\int_{x_{m-k}^*}^{x_{m-k+1}^*} {\ensuremath {\accentset{\triangle}{V}}}_{m-k}(x) \nu_0(dx) &= \frac{1}{2} + \frac{1}{6k} + \frac{1}{24k^2} + O\Big(\frac{1}{k^3}\Big).
\end{aligned}$$ In particular, for $m \gg 0$ and $m-k \leq j_{\max}$, so that also $k \gg 0$, all the integrals $\int_{x_{j-1}^*}^{x_{j+1}^*} {\ensuremath {\accentset{\triangle}{V}}}_j(x) \nu_0(dx)$ are bigger than 1 (it is immediate to see that the same is true for ${\ensuremath {\accentset{\triangle}{V}}}_2$, as well). From now on we will fix a $k \geq \frac{\varepsilon_m m}{H(m)}$ and let $j = m-k$.
Summing together the conditions $\int_I V_i(x)\nu_0(dx)=1$ $\forall i>j$ and noticing that the function $\sum_{i = j}^m V_i$ is constantly equal to $\frac{1}{\mu_m}$ on $[x_j^*,\infty)$ we have: $$\begin{aligned}
\int_{x_{j-1}^*}^{x_j^*} A_j(x) \nu_0(dx) &= m-j+1 - \frac{1}{\mu_m} \nu_0([x_j^*, \infty)) - \int_{x_{j-1}^*}^{x_j^*} {\ensuremath {\accentset{\triangle}{V}}}_j(x) \nu_0(dx)\\
&= k+1- \frac{1}{\log(1+\frac{1}{k})} - \frac{1}{2} + \frac{1}{6k} + O\Big(\frac{1}{k^2}\Big) = \frac{1}{4k} + O\Big(\frac{1}{k^2}\Big)
\end{aligned}$$ Our choice of $A_j$ allows us to compute this integral explicitly: $$\int_{x_{j-1}^*}^{x_j^*} b_j (x-x_{j-1}^*)^2(x-x_j^*)^2 \frac{dx}{x^2} = b_j \big(\varepsilon_m (m-1)\big)^3 \Big(\frac{2}{3} \frac{1}{k^4} + O\Big(\frac{1}{k^5}\Big)\Big).$$ In particular one gets that asymptotically $$b_j \sim \frac{1}{(\varepsilon_m(m-1))^3} \frac{3}{2} k^4 \frac{1}{4k} \sim \bigg(\frac{k}{\varepsilon_m m}\bigg)^3.$$ This immediately allows us to bound the first order derivative of $A_j$ as asked in point 2: Indeed, it is bounded above by $2 b_j \ell_{j-1}^3$ where $\ell_{j-1}$ is again the length of $J_{j-1}$, namely $\ell_j = \frac{\varepsilon_m(m-1)}{k(k+1)} \sim \frac{\varepsilon_m m}{k^2}$. It follows that for $m$ big enough: $$\displaystyle{\sup_{x\in I}|A_j'(x)|} \leq \frac{1}{k^3} \ll \frac{1}{\mu_m(x_j^*-x_{j-1}^*)} \sim \bigg(\frac{k}{\varepsilon_m m}\bigg)^2.$$ The second order derivative of $A_j(x)$ can be easily computed to be bounded by $4 b_j \ell_j^2$. Also remark that the conditions that $|f|$ is bounded by $M$ and that $f'$ is Hölder, say $|f'(x) - f'(y)| \leq K |x-y|^\gamma$, together give a uniform $L_\infty$ bound of $|f'|$ by $2M + K$. Summing up, we obtain: $$|\hat f_m''(x_j^*)| \lesssim b_j \ell_m^3 \mu_m \sim \frac{1}{k^3\varepsilon_m m}$$ (here and in the following we use the symbol $\lesssim$ to stress that we work up to constants and to higher order terms). The leading term of the rest $\hat R_j$ of the Taylor expansion of $\hat f_m$ near $x_j^*$ is $$\hat f_m''(x_j^*) |x-x_j^*|^2 \sim |f_m''(x_j^*)| \ell_j^2 \sim \frac{\varepsilon_m m}{k^7}.$$
Using Lemmas \[lemma:ch4bounds\] and \[lemma:ch4abc\] (taking into consideration Remark \[rmk:nonlinear\]) we obtain $$\begin{aligned}
\int_{\varepsilon_m}^{\infty} |f(x) - \hat f_m(x)|^2 \nu_0(dx) &\lesssim \sum_{j=2}^{j_{\max}} \int_{J_j} |f(x) - \hat f_m(x)|^2 \nu_0(dx) + \int_{H(m)}^\infty |f(x)-\hat f_m(x)|^2 \nu_0(dx) \nonumber \\
&\lesssim \sum_{k=\frac{\varepsilon_m m}{H(m)}}^{m}\mu_m \bigg( \frac{(\varepsilon_m m)^{2+2\gamma}}{k^{4+4\gamma}} + \frac{(\varepsilon_m m)^2}{k^{14}}\bigg) + \frac{1}{H(m)}\sup_{x\geq H(m)}f(x)^2 \label{eq:xquadro} \\
&\lesssim \bigg(\frac{H(m)^{3+4\gamma}}{(\varepsilon_m m)^{2+2\gamma}} + \frac{H(m)^{13}}{(\varepsilon_m m)^{10}}\bigg) + \frac{1}{H(m)}. \nonumber
\end{aligned}$$ It is easy to see that, since $0 < \gamma \leq 1$, as soon as the first term converges, it does so more slowly than the second one. Thus, an optimal choice for $H(m)$ is given by $\sqrt{\varepsilon_m m}$, that gives a rate of convergence: $$L_2(f,\hat f_m)^2 \lesssim \frac{1}{\sqrt{\varepsilon_m m}}.$$ This directly gives a bound on $H(f, \hat f_m)$. Also, the bound on the term $A_m(f)$, which is $L_2(\sqrt f,\widehat{\sqrt{f}}_m)^2$, follows as well, since $f \in {\ensuremath {\mathscr{F}}}_{(\gamma,K,\kappa,M)}^I$ implies $\sqrt{f} \in {\ensuremath {\mathscr{F}}}_{(\gamma, \frac{K}{\sqrt\kappa}, \sqrt \kappa, \sqrt M)}^I$. Finally, the term $B_m^2(f)$ contributes with the same rates as those in : Using Lemma \[lemma:ch4abc\], $$\begin{aligned}
B_m^2(f) &\lesssim \sum_{j=2}^{\lceil m-\frac{\varepsilon_m(m-1)}{H(m)} \rceil} \nu_0(J_j) \|R_j\|_{L_\infty}^2 + \nu_0([H(m), \infty))\\
&\lesssim \mu_m \sum_{k=\frac{\varepsilon_m (m-1)}{H(m)}}^m \Big(\frac{\varepsilon_m m}{k^2}\Big)^{2+2\gamma} + \frac{1}{H(m)}\\
&\lesssim \frac{H(m)^{3+4\gamma}}{(\varepsilon_m m)^{2+2\gamma}} + \frac{1}{H(m)}.
\end{aligned}$$
Proof of Example \[ex:ch4CPP\] {#subsec:ch4ex1}
------------------------------
In this case, since $\varepsilon_m = 0$, the proofs of Theorems \[ch4teo1\] and \[ch4teo2\] simplify and give better estimates near zero, namely: $$\begin{aligned}
\Delta({\ensuremath {\mathscr{P}}}_{n,FV}^{{\ensuremath{\textnormal{Leb}}}}, {\ensuremath {\mathscr{W}}}_n^{\nu_0}) &\leq C_1 \bigg(\sqrt{T_n}\sup_{f\in {\ensuremath {\mathscr{F}}}}\Big(A_m(f)+ B_m(f)+L_2(f,\hat f_m)\Big)+\sqrt{\frac{m^2}{T_n}}\bigg)\nonumber \\
\Delta({\ensuremath {\mathscr{Q}}}_{n,FV}^{{\ensuremath{\textnormal{Leb}}}}, {\ensuremath {\mathscr{W}}}_n^{\nu_0}) &\leq C_2\bigg(\sqrt{n\Delta_n^2}+\frac{m\ln m}{\sqrt{n}}+\sqrt{T_n}\sup_{f\in{\ensuremath {\mathscr{F}}}}\Big( A_m(f)+ B_m(f)+H\big(f,\hat f_m\big)\Big) \bigg) \label{eq:CPP},\end{aligned}$$ where $C_1$, $C_2$ depend only on $\kappa,M$ and $$\begin{aligned}
&A_m(f)=\sqrt{\int_0^1\Big(\widehat{\sqrt f}_m(y)-\sqrt{f(y)}\Big)^2dy},\quad
B_m(f)=\sum_{j=1}^m\bigg(\sqrt m\int_{J_j}\sqrt{f(y)}dy-\sqrt{\theta_j}\bigg)^2.\end{aligned}$$
As a consequence we get: $$\begin{aligned}
\Delta({\ensuremath {\mathscr{P}}}_{n,FV}^{{\ensuremath{\textnormal{Leb}}}},{\ensuremath {\mathscr{W}}}_n^{\nu_0})&\leq O\bigg(\sqrt{T_n}(m^{-\frac{3}{2}}+m^{-1-\gamma})+\sqrt{m^2T_n^{-1}}\bigg).\end{aligned}$$ To get the bounds in the statement of Example \[ex:ch4CPP\] the optimal choices are $m_n = T_n^{\frac{1}{2+\gamma}}$ when $\gamma \leq \frac{1}{2}$ and $m_n = T_n^{\frac{2}{5}}$ otherwise. Concerning the discrete model, we have: $$\begin{aligned}
\Delta({\ensuremath {\mathscr{Q}}}_{n,FV}^{{\ensuremath{\textnormal{Leb}}}},{\ensuremath {\mathscr{W}}}_n^{\nu_0})&\leq O\bigg(\sqrt{n\Delta_n^2}+\frac{m\ln m}{\sqrt{n}}+ \sqrt{n\Delta_n}\big(m^{-\frac{3}{2}}+m^{-1-\gamma}\big)\bigg).\end{aligned}$$ There are four possible scenarios: If $\gamma>\frac{1}{2}$ and $\Delta_n=n^{-\beta}$ with $\frac{1}{2}<\beta<\frac{3}{4}$ (resp. $\beta\geq \frac{3}{4}$) then the optimal choice is $m_n=n^{1-\beta}$ (resp. $m_n=n^{\frac{2-\beta}{5}}$).
If $\gamma\geq\frac{1}{2}$ and $\Delta_n=n^{-\beta}$ with $\frac{1}{2}<\beta<\frac{2+2\gamma}{3+2\gamma}$ (resp. $\beta\geq \frac{2+2\gamma}{3+2\gamma}$) then the optimal choice is $m_n=n^{\frac{2-\beta}{4+2\gamma}}$ (resp. $m_n=n^{1-\beta}$).
Proof of Example \[ch4ex2\] {#subsec:ch4ex2}
---------------------------
As in Examples \[ex:ch4esempi\], we let $\varepsilon_m=m^{-1-\alpha}$ and consider the standard triangular/trapezoidal $V_j$’s. In particular, $\hat f_m$ will be piecewise linear. Condition (C2’) is satisfied and we have $C_m(f)=O(\varepsilon_m)$. This bound, combined with the one obtained in , allows us to conclude that an upper bound for the rate of convergence of $\Delta({\ensuremath {\mathscr{Q}}}_{n,FV}^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0})$ is given by: $$\Delta({\ensuremath {\mathscr{Q}}}_{n,FV}^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0})\leq C \bigg(\sqrt{\sqrt{n^2\Delta_n}\varepsilon_m}+\sqrt{n\Delta_n}\Big(\frac{\ln (\varepsilon_m^{-1})}{m}\Big)^{2}+\frac{m\ln m}{\sqrt n}+\sqrt{n\Delta_n^2}\ln (\varepsilon_m^{-1}) \bigg),$$ where $C$ is a constant only depending on the bound on $\lambda > 0$.
The sequences $\varepsilon_m$ and $m$ can be chosen arbitrarily to optimize the rate of convergence. It is clear from the expression above that, if we take $\varepsilon_m = m^{-1-\alpha}$ with $\alpha > 0$, bigger values of $\alpha$ reduce the first term $\sqrt{\sqrt{n^2\Delta_n}\varepsilon_m}$, while changing the other terms only by constants. It can be seen that taking $\alpha \geq 15$ is enough to make the first term negligeable with respect to the others. In that case, and under the assumption $\Delta_n = n^{-\beta}$, the optimal choice for $m$ is $m = n^\delta$ with $\delta = \frac{5-4\beta}{14}$. In that case, the global rate of convergence is $$\Delta({\ensuremath {\mathscr{Q}}}_{n,FV}^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0}) = \begin{cases}
O\big(n^{\frac{1}{2}-\beta} \ln n\big) & \text{if } \frac{1}{2} < \beta \leq \frac{9}{10}\\
O\big(n^{-\frac{1+2\beta}{7}} \ln n\big) & \text{if } \frac{9}{10} < \beta < 1.
\end{cases}$$
In the same way one can find $$\Delta({\ensuremath {\mathscr{P}}}_{n,FV}^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0})=O\bigg( \sqrt{n\Delta_n} \Big(\frac{\ln m}{m}\Big)^2 \sqrt{\ln(\varepsilon_m^{-1})} + \sqrt{\frac{m^2}{n\Delta_n \ln(\varepsilon_m)}} + \sqrt{n \Delta_n} \varepsilon_m \bigg).$$ As above, we can freely choose $\varepsilon_m$ and $m$ (in a possibly different way from above). Again, as soon as $\varepsilon_m = m^{-1-\alpha}$ with $\alpha \geq 1$ the third term plays no role, so that we can choose $\varepsilon_m = m^{-2}$. Letting $\Delta_n = n^{-\beta}$, $0 < \beta < 1$, and $m = n^\delta$, an optimal choice is $\delta = \frac{1-\beta}{3}$, giving $$\Delta({\ensuremath {\mathscr{P}}}_{n,FV}^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0})=O\Big(n^{\frac{\beta-1}{6}} \big(\ln n\big)^{\frac{5}{2}}\Big) = O\Big(T_n^{-\frac{1}{6}} \big(\ln T_n\big)^\frac{5}{2}\Big).$$
Proof of Example \[ex3\] {#subsec:ch4ex3}
------------------------
Using the computations in , combined with $\big(f(y)-\hat f_m(y)\big)^2\leq 4 \exp(-2\lambda_0 y^3) \leq 4 \exp(-2\lambda_0 H(m)^3)$ for all $y \geq H(m)$, we obtain: $$\begin{aligned}
\int_{\varepsilon_m}^\infty \big|f(x) - \hat f_m(x)\big|^2 \nu_0(dx) &\lesssim \frac{H(m)^{7}}{(\varepsilon_m m)^{4}} + \int_{H(m)}^\infty \big|f(x) - \hat f_m(x)\big|^2 \nu_0(dx)\\
&\lesssim \frac{H(m)^{7}}{(\varepsilon_m m)^{4}} + \frac{e^{-2\lambda_0 H(m)^3}}{H(m)}.
\end{aligned}$$ As in Example \[ex:ch4esempi\], this bounds directly $H^2(f, \hat f_m)$ and $A_m^2(f)$. Again, the first part of the integral appearing in $B_m^2(f)$ is asymptotically smaller than the one appearing above: $$\begin{aligned}
B_m^2(f) &= \sum_{j=1}^m \bigg(\frac{1}{\sqrt{\mu_m}} \int_{J_j} \sqrt{f} \nu_0 - \sqrt{\int_{J_j} f(x) \nu_0(dx)}\bigg)^2\\
&\lesssim \frac{H(m)^{7}}{(\varepsilon_m m)^{4}} + \sum_{k=1}^{\frac{\varepsilon_m m}{H(m)}} \bigg( \frac{1}{\sqrt{\mu_m}} \int_{J_{m-k}} \sqrt{f} \nu_0 - \sqrt{\int_{J_{m-k}} f(x) \nu_0(dx)}\bigg)^2\\
&\lesssim \frac{H(m)^{7}}{(\varepsilon_m m)^{4}} + \frac{e^{-\lambda_0 H(m)^3}}{H(m)}.
\end{aligned}$$ As above, for the last inequality we have bounded $f$ in each $J_{m-k}$, $k \leq \frac{\varepsilon_m m}{H(m)}$, with $\exp(-\lambda_0 H(m)^3)$. Thus the global rate of convergence of $L_2(f,\hat f_m)^2 + A_m^2(f) + B_m^2(f)$ is $\frac{H(m)^{7}}{(\varepsilon_m m)^{4}} + \frac{e^{-\lambda_0 H(m)^3}}{H(m)}$.
Concerning $C_m(f)$, we have $C_m^2(f) = \int_0^{\varepsilon_m} \frac{(\sqrt{f(x)} - 1)^2}{x^2} dx \lesssim \varepsilon_m^5$. To write the global rate of convergence of the Le Cam distance in the discrete setting we make the choice $H(m) = \sqrt[3]{\frac{\eta}{\lambda_0}\ln m}$, for some constant $\eta$, and obtain: $$\begin{aligned}
\Delta({\ensuremath {\mathscr{Q}}}_{n}^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0}) &= O \bigg( \frac{\sqrt{n} \Delta_n}{\varepsilon_m} + \frac{m \ln m}{\sqrt{n}} + \sqrt{n \Delta_n} \Big( \frac{(\ln m)^{\frac{7}{6}}}{(\varepsilon_m m)^2} + \frac{m^{-\frac{\eta}{2}}}{\sqrt[3]{\ln m}} \Big) + \sqrt[4]{n^2 \Delta_n \varepsilon_m^5}\bigg).
\end{aligned}$$ Letting $\Delta_n = n^{-\beta}$, $\varepsilon_m = n^{-\alpha}$ and $m = n^\delta$, optimal choices give $\alpha = \frac{\beta}{3}$ and $\delta = \frac{1}{3}+\frac{\beta}{18}$. We can also take $\eta = 2$ to get a final rate of convergence: $$\Delta({\ensuremath {\mathscr{Q}}}_{n}^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0}) = \begin{cases}
O\big(n^{\frac{1}{2} - \frac{2}{3}\beta}\big)& \text{if } \frac{3}{4} < \beta < \frac{12}{13}\\
O\big(n^{-\frac{1}{6}+\frac{\beta}{18}} (\ln n)^{\frac{7}{6}}\big) &\text{if } \frac{12}{13} \leq \beta < 1.
\end{cases}$$
In the continuous setting, we have $$\Delta({\ensuremath {\mathscr{P}}}_{n}^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0})=O\bigg(\sqrt{n\Delta_n} \Big( \frac{(\ln m)^\frac{7}{6}}{(\varepsilon_m m)^2} + \frac{m^{-\frac{\eta}{2}}}{\sqrt[3]{\ln m}} + \varepsilon_m^{\frac{5}{2}}\Big) + \sqrt{\frac{\varepsilon_m m^2}{n\Delta_n}} \bigg).$$ Using $T_n = n\Delta_n$, $\varepsilon_m = T_n^{-\alpha}$ and $m = T_n^\delta$, optimal choices are given by $\alpha = \frac{4}{17}$, $\delta = \frac{9}{17}$; choosing any $\eta \geq 3$ we get the rate of convergence $$\Delta({\ensuremath {\mathscr{P}}}_{n}^{\nu_0},{\ensuremath {\mathscr{W}}}_n^{\nu_0})=O\big(T_n^{-\frac{3}{34}} (\ln T_n)^{\frac{7}{6}}\big).$$
Background
==========
Le Cam theory of statistical experiments {#sec:ch4lecam}
----------------------------------------
A *statistical model* or *experiment* is a triplet ${\ensuremath {\mathscr{P}}}_j=({\ensuremath {\mathscr{X}}}_j,{\ensuremath {\mathscr{A}}}_j,\{P_{j,\theta}; \theta\in\Theta\})$ where $\{P_{j,\theta}; \theta\in\Theta\}$ is a family of probability distributions all defined on the same $\sigma$-field ${\ensuremath {\mathscr{A}}}_j$ over the *sample space* ${\ensuremath {\mathscr{X}}}_j$ and $\Theta$ is the *parameter space*. The *deficiency* $\delta({\ensuremath {\mathscr{P}}}_1,{\ensuremath {\mathscr{P}}}_2)$ of ${\ensuremath {\mathscr{P}}}_1$ with respect to ${\ensuremath {\mathscr{P}}}_2$ quantifies “how much information we lose” by using ${\ensuremath {\mathscr{P}}}_1$ instead of ${\ensuremath {\mathscr{P}}}_2$ and it is defined as $\delta({\ensuremath {\mathscr{P}}}_1,{\ensuremath {\mathscr{P}}}_2)=\inf_K\sup_{\theta\in \Theta}||KP_{1,\theta}-P_{2,\theta}||_{TV},$ where TV stands for “total variation” and the infimum is taken over all “transitions” $K$ (see [@lecam], page 18). The general definition of transition is quite involved but, for our purposes, it is enough to know that Markov kernels are special cases of transitions. By $KP_{1,\theta}$ we mean the image measure of $P_{1,\theta}$ via the Markov kernel $K$, that is $$KP_{1,\theta}(A)=\int_{{\ensuremath {\mathscr{X}}}_1}K(x,A)P_{1,\theta}(dx),\quad\forall A\in {\ensuremath {\mathscr{A}}}_2.$$ The experiment $K{\ensuremath {\mathscr{P}}}_1=({\ensuremath {\mathscr{X}}}_2,{\ensuremath {\mathscr{A}}}_2,\{KP_{1,\theta}; \theta\in\Theta\})$ is called a *randomization* of ${\ensuremath {\mathscr{P}}}_1$ by the Markov kernel $K$. When the kernel $K$ is deterministic, that is $K(x,A)={\ensuremath {\mathbb{I}}}_{A}S(x)$ for some random variable $S:({\ensuremath {\mathscr{X}}}_1,{\ensuremath {\mathscr{A}}}_1)\to({\ensuremath {\mathscr{X}}}_2,{\ensuremath {\mathscr{A}}}_2)$, the experiment $K{\ensuremath {\mathscr{P}}}_1$ is called the *image experiment by the random variable* $S$. The Le Cam distance is defined as the symmetrization of $\delta$ and it defines a pseudometric. When $\Delta({\ensuremath {\mathscr{P}}}_1,{\ensuremath {\mathscr{P}}}_2)=0$ the two statistical models are said to be *equivalent*. Two sequences of statistical models $({\ensuremath {\mathscr{P}}}_{1}^n)_{n\in{\ensuremath {\mathbb{N}}}}$ and $({\ensuremath {\mathscr{P}}}_{2}^n)_{n\in{\ensuremath {\mathbb{N}}}}$ are called *asymptotically equivalent* if $\Delta({\ensuremath {\mathscr{P}}}_{1}^n,{\ensuremath {\mathscr{P}}}_{2}^n)$ tends to zero as $n$ goes to infinity. A very interesting feature of the Le Cam distance is that it can be also translated in terms of statistical decision theory. Let ${\ensuremath {\mathscr{D}}}$ be any (measurable) decision space and let $L:\Theta\times {\ensuremath {\mathscr{D}}}\mapsto[0,\infty)$ denote a loss function. Let $\|L\|=\sup_{(\theta,z)\in\Theta\times{\ensuremath {\mathscr{D}}}}L(\theta,z)$. Let $\pi_i$ denote a (randomized) decision procedure in the $i$-th experiment. Denote by $R_i(\pi_i,L,\theta)$ the risk from using procedure $\pi_i$ when $L$ is the loss function and $\theta$ is the true value of the parameter. Then, an equivalent definition of the deficiency is: $$\begin{aligned}
\delta({\ensuremath {\mathscr{P}}}_1,{\ensuremath {\mathscr{P}}}_2)=\inf_{\pi_1}\sup_{\pi_2}\sup_{\theta\in\Theta}\sup_{L:\|L\|=1}\big|R_1(\pi_1,L,\theta)-R_2(\pi_2,L,\theta)\big|.\end{aligned}$$ Thus $\Delta({\ensuremath {\mathscr{P}}}_1,{\ensuremath {\mathscr{P}}}_2)<\varepsilon$ means that for every procedure $\pi_i$ in problem $i$ there is a procedure $\pi_j$ in problem $j$, $\{i,j\}=\{1,2\}$, with risks differing by at most $\varepsilon$, uniformly over all bounded $L$ and $\theta\in\Theta$. In particular, when minimax rates of convergence in a nonparametric estimation problem are obtained in one experiment, the same rates automatically hold in any asymptotically equivalent experiment. There is more: When explicit transformations from one experiment to another are obtained, statistical procedures can be carried over from one experiment to the other one.
There are various techniques to bound the Le Cam distance. We report below only the properties that are useful for our purposes. For the proofs see, e.g., [@lecam; @strasser].
\[ch4delta0\] Let ${\ensuremath {\mathscr{P}}}_j=({\ensuremath {\mathscr{X}}},{\ensuremath {\mathscr{A}}},\{P_{j,\theta}; \theta\in\Theta\})$, $j=1,2$, be two statistical models having the same sample space and define $\Delta_0({\ensuremath {\mathscr{P}}}_1,{\ensuremath {\mathscr{P}}}_2):=\sup_{\theta\in\Theta}\|P_{1,\theta}-P_{2,\theta}\|_{TV}.$ Then, $\Delta({\ensuremath {\mathscr{P}}}_1,{\ensuremath {\mathscr{P}}}_2)\leq \Delta_0({\ensuremath {\mathscr{P}}}_1,{\ensuremath {\mathscr{P}}}_2)$.
In particular, Property \[ch4delta0\] allows us to bound the Le Cam distance between statistical models sharing the same sample space by means of classical bounds for the total variation distance. To that aim, we collect below some useful results.
\[ch4h\] Let $P_1$ and $P_2$ be two probability measures on ${\ensuremath {\mathscr{X}}}$, dominated by a common measure $\xi$, with densities $g_{i}=\frac{dP_{i}}{d\xi}$, $i=1,2$. Define $$\begin{aligned}
L_1(P_1,P_2)&=\int_{{\ensuremath {\mathscr{X}}}} |g_{1}(x)-g_{2}(x)|\xi(dx), \\
H(P_1,P_2)&=\bigg(\int_{{\ensuremath {\mathscr{X}}}} \Big(\sqrt{g_{1}(x)}-\sqrt{g_{2}(x)}\Big)^2\xi(dx)\bigg)^{1/2}.
\end{aligned}$$ Then, $$\|P_1-P_2\|_{TV}=\frac{1}{2}L_1(P_1,P_2)\leq H(P_1,P_2).$$
\[ch4hp\] Let $P$ and $Q$ be two product measures defined on the same sample space: $P=\otimes_{i=1}^n P_i$, $Q=\otimes_{i=1}^n Q_i$. Then $$H ^2(P,Q)\leq \sum_{i=1}^nH^2(P_i,Q_i).$$
\[fact:ch4hellingerpoisson\] Let $P_i$, $i=1,2$, be the law of a Poisson random variable with mean $\lambda_i$. Then $$H^2(P_1,P_2)=1-\exp\bigg(-\frac{1}{2}\Big(\sqrt{\lambda_1}-\sqrt{\lambda_2}\Big)^2\bigg).$$
\[fact:ch4gaussiane\] Let $Q_1\sim{\ensuremath {\mathscr{Nn}}}(\mu_1,\sigma_1^2)$ and $Q_2\sim{\ensuremath {\mathscr{Nn}}}(\mu_2,\sigma_2^2)$. Then $$\|Q_1-Q_2\|_{TV}\leq \sqrt{2\bigg(1-\frac{\sigma_1^2}{\sigma_2^2}\bigg)^2+\frac{(\mu_1-\mu_2)^2}{2\sigma_2^2}}.$$
\[fact:ch4processigaussiani\] For $i=1,2$, let $Q_i$, $i=1,2$, be the law on $(C,{\ensuremath {\mathscr{C}}})$ of two Gaussian processes of the form $$X^i_t=\int_{0}^t h_i(s)ds+ \int_0^t \sigma(s)dW_s,\ t\in[0,T]$$ where $h_i\in L_2({\ensuremath {\mathbb{R}}})$ and $\sigma\in{\ensuremath {\mathbb{R}}}_{>0}$. Then: $$L_1\big(Q_1,Q_2\big)\leq \sqrt{\int_{0}^T\frac{\big(h_1(y)-h_2(y)\big)^2}{\sigma^2(s)}ds}.$$
\[ch4fatto3\] Let ${\ensuremath {\mathscr{P}}}_i=({\ensuremath {\mathscr{X}}}_i,{\ensuremath {\mathscr{A}}}_i,\{P_{i,\theta}, \theta\in\Theta\})$, $i=1,2$, be two statistical models. Let $S:{\ensuremath {\mathscr{X}}}_1\to{\ensuremath {\mathscr{X}}}_2$ be a sufficient statistics such that the distribution of $S$ under $P_{1,\theta}$ is equal to $P_{2,\theta}$. Then $\Delta({\ensuremath {\mathscr{P}}}_1,{\ensuremath {\mathscr{P}}}_2)=0$.
\[ch4independentkernels\] Let $P_i$ be a probability measure on $(E_i,\mathcal{E}_i)$ and $K_i$ a Markov kernel on $(G_i,\mathcal G_i)$. One can then define a Markov kernel $K$ on $(\prod_{i=1}^n E_i,\otimes_{i=1}^n \mathcal{G}_i)$ in the following way: $$K(x_1,\dots,x_n; A_1\times\dots\times A_n):=\prod_{i=1}^nK_i(x_i,A_i),\quad \forall x_i\in E_i,\ \forall A_i\in \mathcal{G}_i.$$ Clearly $K\otimes_{i=1}^nP_i=\otimes_{i=1}^nK_iP_i$.
Finally, we recall the following result that allows us to bound the Le Cam distance between Poisson and Gaussian variables.
\[ch4teomisto\](See [@BC04], Theorem 4) Let $\tilde P_{\lambda}$ be the law of a Poisson random variable $\tilde X_{\lambda}$ with mean $\lambda$. Furthermore, let $P_{\lambda}^*$ be the law of a random variable $Z^*_{\lambda}$ with Gaussian distribution ${\ensuremath {\mathscr{Nn}}}(2\sqrt{\lambda},1)$, and let $\tilde U$ be a uniform variable on $\big[-\frac{1}{2},\frac{1}{2}\big)$ independent of $\tilde X_{\lambda}$. Define $$\tilde Z_{\lambda}=2\textnormal{sgn}\big(\tilde X_{\lambda}+\tilde U\big)\sqrt{\big|\tilde X_{\lambda}+\tilde U\big|}.$$ Then, denoting by $P_{\lambda}$ the law of $\tilde Z_{\lambda}$, $$H ^2\big(P_{\lambda}, P_{\lambda}^*\big)=O(\lambda^{-1}).$$
Thanks to Theorem \[ch4teomisto\], denoting by $\Lambda$ a subset of ${\ensuremath {\mathbb{R}}}_{>0}$, by $\tilde {\ensuremath {\mathscr{P}}}$ (resp. ${\ensuremath {\mathscr{P}}}^*$) the statistical model associated with the family of probabilities $\{\tilde P_\lambda: \lambda \in \Lambda\}$ (resp. $\{P_\lambda^* : \lambda \in \Lambda\}$), we have $$\Delta\big(\tilde {\ensuremath {\mathscr{P}}}, {\ensuremath {\mathscr{P}}}^*\big) \leq \sup_{\lambda \in \Lambda} \frac{C}{\lambda},$$ for some constant $C$. Indeed, the correspondence associating $\tilde Z_\lambda$ to $\tilde X_\lambda$ defines a Markov kernel; conversely, associating to $\tilde Z_\lambda$ the closest integer to its square, defines a Markov kernel going in the other direction.
Lévy processes {#sec:ch4levy}
--------------
A stochastic process $\{X_t:t\geq 0\}$ on ${\ensuremath {\mathbb{R}}}$ defined on a probability space $(\Omega,{\ensuremath {\mathscr{A}}},{\ensuremath {\mathbb{P}}})$ is called a *Lévy process* if the following conditions are satisfied.
1. $X_0=0$ ${\ensuremath {\mathbb{P}}}$-a.s.
2. For any choice of $n\geq 1$ and $0\leq t_0<t_1<\ldots<t_n$, random variables $X_{t_0}$, $X_{t_1}-X_{t_0},\dots ,X_{t_n}-X_{t_{n-1}}$are independent.
3. The distribution of $X_{s+t}-X_s$ does not depend on $s$.
4. There is $\Omega_0\in {\ensuremath {\mathscr{A}}}$ with ${\ensuremath {\mathbb{P}}}(\Omega_0)=1$ such that, for every $\omega\in \Omega_0$, $X_t(\omega)$ is right-continuous in $t\geq 0$ and has left limits in $t>0$.
5. It is stochastically continuous.
Thanks to the *Lévy-Khintchine formula*, the characteristic function of any Lévy process $\{X_t\}$ can be expressed, for all $u$ in ${\ensuremath {\mathbb{R}}}$, as: $$\label{caratteristica}
{\ensuremath {\mathbb{E}}}\big[e^{iuX_t}\big]=\exp\bigg(-t\Big(iub-\frac{u^2\sigma^2}{2}-\int_{{\ensuremath {\mathbb{R}}}}(1-e^{iuy}+iuy{\ensuremath {\mathbb{I}}}_{\vert y\vert \leq 1})\nu(dy)\Big)\bigg),$$ where $b,\sigma\in {\ensuremath {\mathbb{R}}}$ and $\nu$ is a measure on ${\ensuremath {\mathbb{R}}}$ satisfying $$\nu(\{0\})=0 \textnormal{ and } \int_{{\ensuremath {\mathbb{R}}}}(|y|^2\wedge 1)\nu(dy)<\infty.$$ In the sequel we shall refer to $(b,\sigma^2,\nu)$ as the characteristic triplet of the process $\{X_t\}$ and $\nu$ will be called the *Lévy measure*. This data characterizes uniquely the law of the process $\{X_t\}$.
Let $D=D([0,\infty),{\ensuremath {\mathbb{R}}})$ be the space of mappings $\omega$ from $[0,\infty)$ into ${\ensuremath {\mathbb{R}}}$ that are right-continuous with left limits. Define the *canonical process* $x:D\to D$ by $$\forall \omega\in D,\quad x_t(\omega)=\omega_t,\;\;\forall t\geq 0.$$
Let ${\ensuremath {\mathscr{D}}}_t$ and ${\ensuremath {\mathscr{D}}}$ be the $\sigma$-algebras generated by $\{x_s:0\leq s\leq t\}$ and $\{x_s:0\leq s<\infty\}$, respectively (here, we use the same notations as in [@sato]).
By the condition (4) above, any Lévy process on ${\ensuremath {\mathbb{R}}}$ induces a probability measure $P$ on $(D,{\ensuremath {\mathscr{D}}})$. Thus $\{X_t\}$ on the probability space $(D,{\ensuremath {\mathscr{D}}},P)$ is identical in law with the original Lévy process. By saying that $(\{x_t\},P)$ is a Lévy process, we mean that $\{x_t:t\geq 0\}$ is a Lévy process under the probability measure $P$ on $(D,{\ensuremath {\mathscr{D}}})$. For all $t>0$ we will denote $P_t$ for the restriction of $P$ to ${\ensuremath {\mathscr{D}}}_t$. In the case where $\int_{|y|\leq 1}|y|\nu(dy)<\infty$, we set $\gamma^{\nu}:=\int_{|y|\leq 1}y\nu(dy)$. Note that, if $\nu$ is a finite Lévy measure, then the process having characteristic triplet $(\gamma^{\nu},0,\nu)$ is a compound Poisson process.
Here and in the sequel we will denote by $\Delta x_r$ the jump of process $\{x_t\}$ at the time $r$: $$\Delta x_r = x_r - \lim_{s \uparrow r} x_s.$$ For the proof of Theorems \[ch4teo1\], \[ch4teo2\] we also need some results on the equivalence of measures for Lévy processes. By the notation $\ll$ we will mean “is absolutely continuous with respect to”.
\[ch4teosato\] Let $P^1$ (resp. $P^2$) be the law induced on $(D,{\ensuremath {\mathscr{D}}})$ by a Lévy process of characteristic triplet $(\eta,0,\nu_1)$ (resp. $(0,0,\nu_2)$), where $$\label{ch4gamma*}
\eta=\int_{\vert y \vert \leq 1}y(\nu_1-\nu_2)(dy)$$ is supposed to be finite. Then $P_t^1\ll P_t^2$ for all $t\geq 0$ if and only if $\nu_1\ll\nu_2$ and the density $\frac{d\nu_1}{d\nu_2}$ satisfies $$\label{ch4Sato}
\int\bigg(\sqrt{\frac{d\nu_1}{d\nu_2}(y)}-1\bigg)^2\nu_2(dy)<\infty.$$ Remark that the finiteness in implies that in . When $P_t^1\ll P_t^2$, the density is $$\frac{dP_t^1}{dP_t^2}(x)=\exp(U_t(x)),$$ with $$\label{ch4U}
U_t(x)=\lim_{\varepsilon\to 0} \bigg(\sum_{r\leq t}\ln \frac{d\nu_1}{d\nu_2}(\Delta x_r){\ensuremath {\mathbb{I}}}_{\vert\Delta x_r\vert>\varepsilon}-
\int_{\vert y\vert > \varepsilon} t\bigg(\frac{d\nu_1}{d\nu_2}(y)-1\bigg)\nu_2(dy)\bigg),\\ P^{(0,0,\nu_2)}\textnormal{-a.s.}$$ The convergence in is uniform in $t$ on any bounded interval, $P^{(0,0,\nu_2)}$-a.s. Besides, $\{U_t(x)\}$ defined by is a Lévy process satisfying ${\ensuremath {\mathbb{E}}}_{P^{(0,0,\nu_2)}}[e^{U_t(x)}]=1$, $\forall t\geq 0$.
Finally, let us consider the following result giving an explicit bound for the $L_1$ and the Hellinger distances between two Lévy processes of characteristic triplets of the form $(b_i,0,\nu_i)$, $i=1,2$ with $b_1-b_2=\int_{\vert y \vert \leq 1}y(\nu_1-\nu_2)(dy)$.
\[teo:ch4bound\] For any $0<T<\infty$, let $P_T^i$ be the probability measure induced on $(D,{\ensuremath {\mathscr{D}}}_T)$ by a Lévy process of characteristic triplet $(b_i,0,\nu_i)$, $i=1,2$ and suppose that $\nu_1\ll\nu_2$.
If $H^2(\nu_1,\nu_2):=\int\big(\sqrt{\frac{d\nu_1}{d\nu_2}(y)}-1\big)^2\nu_2(dy)<\infty,$ then $$H^2(P_T^1,P_T^2)\leq \frac{T}{2}H^2(\nu_1,\nu_2).$$
We conclude the Appendix with a technical statement about the Le Cam distance for finite variation models.
\[ch4LC\] $$\Delta({\ensuremath {\mathscr{P}}}_n^{\nu_0},{\ensuremath {\mathscr{P}}}_{n,FV}^{\nu_0})=0.$$
Consider the Markov kernels $\pi_1$, $\pi_2$ defined as follows $$\pi_1(x,A)={\ensuremath {\mathbb{I}}}_{A}(x^d), \quad
\pi_2(x,A)={\ensuremath {\mathbb{I}}}_{A}(x-\cdot \gamma^{\nu_0}),
\quad \forall x\in D, A \in {\ensuremath {\mathscr{D}}},$$ where we have denoted by $x^d$ the discontinuous part of the trajectory $x$, i.e. $\Delta x_r = x_r - \lim_{s \uparrow r} x_s,\ x_t^d=\sum_{r \leq t}\Delta x_r$ and by $x-\cdot \gamma^{\nu_0}$ the trajectory $x_t-t\gamma{\nu_0}$, $t\in[0,T_n]$. On the one hand we have: $$\begin{aligned}
\pi_1 P^{(\gamma^{\nu-\nu_0},0,\nu)}(A)&=\int_D \pi_1(x,A)P^{(\gamma^{\nu-\nu_0},0,\nu)}(dx)=\int_D {\ensuremath {\mathbb{I}}}_A(x^d)P^{(\gamma^{\nu-\nu_0},0,\nu)}(dx)\\
&=P^{(\gamma^{\nu},0,\nu)}(A),\end{aligned}$$ where in the last equality we have used the fact that, under $P^{(\gamma^{\nu-\nu_0},0,\nu)}$, $\{x_t^d\}$ is a Lévy process with characteristic triplet $(\gamma^{\nu},0,\nu)$ (see [@sato], Theorem 19.3). On the other hand: $$\begin{aligned}
\pi_2 P^{(\gamma^{\nu},0,\nu)}(A)&=\int_D \pi_2(x,A)P^{(\gamma^{\nu_0},0,\nu)}(dx)=\int_D {\ensuremath {\mathbb{I}}}_A(x-\cdot \gamma^{\nu_0})P^{(\gamma^{\nu},0,\nu)}(dx)\\
&=P^{(\gamma^{\nu-\nu_0},0,\nu)}(A),\end{aligned}$$ since, by definition, $\gamma^{\nu}-\gamma^{\nu_0}$ is equal to $\gamma^{\nu-\nu_0}$. The conclusion follows by the definition of the Le Cam distance.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I am very grateful to Markus Reiss for several interesting discussions and many insights; this paper would never have existed in the present form without his advice and encouragement. My deepest thanks go to the anonymous referee, whose insightful comments have greatly improved the exposition of the paper; some gaps in the proofs have been corrected thanks to his/her remarks.
|
Moss (Physcomitrella patens) GH3 proteins act in auxin homeostasis.
Auxins are hormones involved in many cellular, physiological and developmental processes in seed plants and in mosses such as Physcomitrella patens. Control of auxin levels is achieved in higher plants via synthesis of auxin conjugates by members of the GH3 family. The role of the two GH3-like proteins from P. patens for growth and auxin homeostasis was therefore analysed. The in vivo-function of the two P. patens GH3 genes was investigated using single and double knockout mutants. The two P. patens GH3 proteins were also heterologously expressed to determine their enzymatic activity. Both P. patens GH3 enzymes accepted the auxin indole acetic acid (IAA) as substrate, but with different preferences for the amino acid to which it is attached. Cytoplasmic localization was shown for PpGH3-1 tagged with green fluorescent protein (GFP). Targeted knock-out of either gene exhibited an increased sensitivity to auxin, resulting in growth inhibition. On plain mineral media mutants had higher levels of free IAA and less conjugated IAA than the wild type, and this effect was enhanced when auxin was supplied. The DeltaPpGH3-1/DeltaPpGH3-2 double knockout had almost no IAA amide conjugates but still synthesized ester conjugates. Taken together, these data suggest a developmentally controlled involvement of P. patens GH3 proteins in auxin homeostasis by conjugating excess of physiologically active free auxin to inactive IAA-amide conjugates. |
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Summer Flowers at Danckerts
Summer is now well and truly on its way now as we come upon another Bank Holiday this weekend.
We have some lovely gardens plants and pots at the shop, as well as a new range of "Vivid Arts" garden animals on display, which are a fantastically realistic range of life size animals and birds to enhance the garden...from frogs to foxes, and rabbits to robins, pop in and take a look!
The gardens in Wednesbury are going to be coming alive with plants, animals, and barbies! The summer flower collection is now in full swing, with some delightful bouquets and vases full of Snaps, Sweet Williams, and other summer favourites.
Keep in touch via Facebook, and we'll keep you notified of any Special Offers that are coming up!
We recently had St Georges day, and the St Georges Day March was hugely popular, starting at Stone Cross, just past the Wednesbury/ West Bromwich border, and finishing up at Dartmouth Park in the Sandwell Valley. |
From 1 July 2018, the Tax Office is advising Australians that if they find an error in their tax return or activity statement they will not incur a penalty but will advise of the error and how to get it right next time.
Penalty relief will only apply to eligible taxpayers or entities (i.e., turnover of less than $10 million) every three years.
Eligible individuals will only be given penalty relief on their tax return or activity statement if they make an inadvertent error because they either:
– took a position on income tax that is not reasonably arguable, or
– failed to take reasonable care
The ATO will not provide penalty relief when individuals have (in the past three years):
Received penalty relief
– Avoided tax payment or committed fraud
– Accrued taxation debts with no intention of being able to pay (i.e., phoenix activity)
– Previously penalised for reckless or intentional disregard of the law
– Participated in the management or control of another entity which has evaded tax.
Individuals can not apply for penalty relief. The ATO is reminding individuals that they will provide relief during an audit should it apply.
Penalty relief will not be applied to:
– Wealthy individuals and their businesses
– Associates of wealthy individuals (that may be deemed a small business entity in their own right)
– Public groups, significant global entities and associates
Penalty relief will also not be applied to certain taxes, i.e., fringe benefits tax (FBT) or super guarantee (SG). |
Q:
How to change XML from dataset into HTML UL
I'm working on a C# webforms application and have a datalayer which gathers information about the menu a customer can see, based on their customer number and order type.
I was using the ASP.NET menu control for this until the qa department asked to change the menu to expand on click instead of hover. At that point, I decided to try and do the menu with a simpler css/html/jquery approach but I've hit a jam.
I have the following method in my datalayer that gets information for the menu and returns it as XML. What I'm stuck on is how to take the XML that was being gathered, when I was using the menu control and hopefully reformat it into a UL for using in the html/css approach I'd like to do.
public static string BuildMenu(string cprcstnm, string docType)
{
DataSet ds = new DataSet();
string connStr = ConfigurationManager.ConnectionStrings["DynamicsConnectionString"].ConnectionString;
using (SqlConnection conn = new SqlConnection(connStr))
{
string sql = "usp_SelectItemMenuByCustomer";
SqlDataAdapter da = new SqlDataAdapter(sql, conn);
da.SelectCommand.CommandType = CommandType.StoredProcedure;
da.SelectCommand.Parameters.Add("@CPRCSTNM", SqlDbType.VarChar).Value = cprcstnm;
da.SelectCommand.Parameters.Add("@DOCID", SqlDbType.VarChar).Value = docType;
da.Fill(ds);
da.Dispose();
}
ds.DataSetName = "Menus";
ds.Tables[0].TableName = "Menu";
DataRelation relation = new DataRelation("ParentChild",
ds.Tables["Menu"].Columns["MenuID"],
ds.Tables["Menu"].Columns["ParentID"],
false);
relation.Nested = true;
ds.Relations.Add(relation);
return ds.GetXml();
}
A sample of XMl that is output is as follows:
<Menus>
- <Menu>
<MenuID>23</MenuID>
<ITEMNMBR>0</ITEMNMBR>
<Text>ACC</Text>
<Description>ACC</Description>
<ParentID>0</ParentID>
- <Menu>
<MenuID>34</MenuID>
<ITEMNMBR>1</ITEMNMBR>
<Text>BASE</Text>
<Description>BASE</Description>
<ParentID>23</ParentID>
- <Menu>
<MenuID>516</MenuID>
<ITEMNMBR>2</ITEMNMBR>
<Text>HYP</Text>
<Description>HYP</Description>
<ParentID>34</ParentID>
I would need to convert this to something such as :
<ul class="dropdown">
<li><a href="#">ACC</a>
<ul class="sub_menu">
<li>
<a href="#">BASE</a>
<ul>
<li>
<a href="#">HYP</a>
<ul>
<li><a href="#">Terminal 1</a></li>
<li><a href="#">Terminal 1</a></li>
</ul>
</li>
</ul>
</li>
A:
You will get some ideas from the following MSDN link that illustrates writing html from a dataset using xslt
http://msdn.microsoft.com/en-us/library/8fd7xytc(v=vs.80).aspx
|
[Central venous dialysis catheter. Silicone rubber dialysis catheter used for permanent vascular access].
51 dual lumen jugularis dialysis catheters (Permcath, Quinton) were placed by surgical technique in 34 patients and by percutaneous technique in eight patients. Mean catheter life-time was 4.1 months. Seven catheters were removed due to complications (infection three catheters, clotting four catheters). Minor flow problems occurred in 8.8% of all procedures. Seven occluded catheters were successfully reopened by use of locally applied streptokinase. A strict aseptic technique is essential to avoid infection. Permcath is an acceptable vascular access device for patients in whom it is impossible to create an arterio-venous fistula. |
Marine Air Control Group 38
Marine Air Control Group 38 (MACG-38) is a United States Marine Corps aviation command and control unit based at Marine Corps Air Station Miramar that is currently composed of five squadrons and one battalion that provide the 3rd Marine Aircraft Wing's tactical headquarters, positive and procedural control to aircraft, and air defense support for the I Marine Expeditionary Force.
Mission
Subordinate units
3rd Low Altitude Air Defense Battalion
Marine Air Control Squadron 1
Marine Air Support Squadron 3
Marine Tactical Air Command Squadron 38
Marine Wing Communications Squadron 38
History
Marine Air Control Group 38 was activated on September 1, 1967 at Marine Corps Air Station El Toro, California. The Group deployed to Saudi Arabia in August 1990 and later supported Operation Desert Storm. Elements of the group have supported Operation Restore Hope, Operation Safe Departure, Operation Southern Watch and Operation Stabilise. The group relocated to MCAS Miramar in October 1998. MACG-38 units began deploying to Kuwait in 2002 and the entire control group would eventually take part in the 2003 invasion of Iraq and continued to deploy today in support of Operation Iraqi Freedom through early 2009. They were headquartered at Al Asad Airbase in the Al Anbar Province from 2004 through the end of their last Iraq deployment in early 2009.
Most recently the Group deployed to Camp Leatherneck, Afghanistan in March 2010. They are responsible for providing aviation command and control for the I Marine Expeditionary Force (I MEF) in support of Operation Enduring Freedom. They returned to The United States in Spring of 2011.
See also
United States Marine Corps Aviation
List of United States Marine Corps aircraft groups
List of United States Marine Corps aircraft squadrons
References
External links
Category:United States Marine Corps air control groups
Category:Military units and formations in California |
The verbals: sports quotes of 1994
There are no small accidents on this circuit. Ayrton Senna, before the San Marino Grand Prix, during which he suffered a fatal crash.
One of my best friends has been killed on the curve where I escaped death. I was lucky; he wasn't. It's like having a cheque book. You start pulling out the pages until one day no pages are left. He was the one driver so perfect nobody thought anything could happen to him. Gerhard Berger, Formula 1 driver, on Ayrton Senna.
It was at the bottom of our hearts to dedicate this victory to our great friend, Ayrton Senna. He was also heading for his fourth title. Claudio Taffarel, Brazil's goalkeeper, following victory in the World Cup final.
There will never be another Senna. The poet of speed is dead. El Diario, Bolivian sports newspaper.
Senna was the greatest driver ever and when someone like him is killed you have to ask yourself what is the point of it all. Nikki Lauda.
When I saw him crash and realised there was no way he was going to be able to continue the race, I cheered with joy. I thought: `He'll be home earlier tonight'. Adrienne Galisteu, Senna's girlfriend. |
**B Grade** CNPS12X Ultimate Performance Triple Fan CPU Cooler
Below is the original description for this product, any reference to warranty is to be ignored. Warranty for this item is 90 days as with all B Grade items.
B Grade items may have been used, have damaged packaging, missing accessories or a combination of these.
Some items may have scuff marks or slight scratches but should otherwise be an operable product.
Renowned for producing some of the world best CPU coolers, Zalman have now released their newest flagship cooler, the CNPS12X. It is the world's first "out of the box" triple fan cooler and is compatible with Intel latest LGA2011 Sandy Bridge E processors.
Worlds first "out of the box" triple fan CPU coolerThere are many CPU coolers available on the market that can accommodate three fans, but to make this happen at least one additional fan needs to be purchased which add to the expense. With the Zalman CNPS12X you get three 120mm blue LED fans built into the cooler so there is no extra costs. Also all three fans run off one fan header, making powering the fans extremely easy.
Six W-DTH composite heatpipes for excellent heat transferFirst seen on the CNPS11X, composite heatpipes help transfer the heat from the CPU up to 50% faster than standard heatpipes. This helps to increase the performance of the cooler even further. The six heatpipes are U-shaped, which effectively double the heat transfer compared to none U-shaped heatpipes.At the base of the cooler (where the heatpipes make contact with the CPU) the heatpipes utilise what Zalman call Whole-Direct Touch Heatpipes (W-DTH). This allows the heatpipes to make direct contact with the CPU, another feature to help increase performance. But not only that, the area of the Direct Touch will cover the whole CPU. Even the new Intel CPUs for LGA2011 will also be covered by W-DTH.
100% nickel plated with blue LED fans for amazing aestheticsMost CPU coolers are hidden inside the computer case where they go about their business unseen. But if you like to show off the internals of the PC you may want a CPU cooler than looks the part, and boy the CNPS12X does look the part!The entire heatsink of CNPS12X is plated with "Black-Pearl" nickel for a long-term corrosion resistance, while the deep "Black-Pearl" tone, along with the high intensity from the blue LED fans helps this cooler stand head and shoulders above the rest. |
Further studies on hepatitis C virus NS5A-SH3 domain interactions: identification of residues critical for binding and implications for viral RNA replication and modulation of cell signalling.
The NS5A protein of hepatitis C virus has been shown to interact with a subset of Src homology 3 (SH3) domain-containing proteins. The molecular mechanisms underlying these observations have not been fully characterized, therefore a previous analysis of NS5A-SH3 domain interactions was extended. By using a semi-quantitative ELISA assay, a hierarchy of binding between various SH3 domains for NS5A was demonstrated. Molecular modelling of a polyproline motif within NS5A (termed PP2.2) bound to the FynSH3 domain predicted that the specificity-determining RT-loop region within the SH3 domain did not interact directly with the PP2.2 motif. However, it was demonstrated that the RT loop did contribute to the specificity of binding, implicating the involvement of other intermolecular contacts between NS5A and SH3 domains. The modelling analysis also predicted a critical role for a conserved arginine located at the C terminus of the PP2.2 motif; this was confirmed experimentally. Finally, it was demonstrated that, in comparison with wild-type replicon cells, inhibition of the transcription factor AP-1, a function previously assigned to NS5A, was not observed in cells harbouring a subgenomic replicon containing a mutation within the PP2.2 motif. However, the ability of the mutated replicon to establish itself within Huh-7 cells was unaffected. The highly conserved nature of the PP2.2 motif within NS5A suggests that functions involving this motif are of importance, but are unlikely to play a role in replication of the viral RNA genome. It is more likely that they play a role in altering the cellular environment to favour viral persistence. |
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Fill in any missing fields and make sure to click ‘Save Changes’ when you are finished. |
You can make an appointment to meet with your Financial Aid counselor using Orange Success through MySlice. Once logged in, select 'Orange SUccess' under 'Advising' in the Student Services panel. Within your Orange SUccess portal, navigate to 'My Success Network,' select your financial aid advisor and schedule an appointment at a day and time convenient for you.
If your counselor is not available at a time that suits your schedule, please call or visit our office to schedule an appointment with the next available counselor. |
Every industry has its own characteristics and requirements. For detailed benefits of our systems related to your industry please make a selection in the left menu. General benefits of using Hitec Power Protection rotary UPS systems are:
Most reliable systemThe simple design has fewer components than for example static UPS systems. This highly improves the reliability (MTBF). Our systems have a lifetime expectancy of more than 25 years.
Most cost and energy efficient systemOperating efficiency of our systems can exceed 97%, because they do not require power conversion in the power path or a conditioned, energy consuming battery room during operation. You also do not need battery replacement every 3 to 5 years, resulting in a lower total cost of ownership (TCO) compared to for example static UPS technologies.
Most environmental friendly systemOur rotary systems have high energy efficiency and do not use batteries. Static UPS systems for example produce a considerable amount of chemical waste during its lifetime, because batteries need to be replaced every 3 to 5 years. Click here to find out more about the environmental benefits of our systems.
Most space efficient systemA static UPS system requires a diesel generator set, power electronics, batteries and numerous auxiliary equipment. Our compact and simple diesel rotary UPS design combines all these components in one, reducing the footprint with 40 up to 60%. |
Q:
how to get result with cursor and paging using ZSCAN command with stackexchange.redis library?
I am using stackexchange.redis.
in that zscan is giving all matched value
I want to get exactly given page size result and next cursor for remaining values.
I have debugged its source code library in that i found that they are
scanning entire source value until cursor became zero and provides all
matched values.
so could we can get result as per cursor same as redis command Zscan.
here is my code snap
using (ConnectionMultiplexer conn = ConnectionMultiplexer.Connect(conf))
{
var dbs = conn.GetDatabase();
int currentpage = 0,pagesize=20;
var scanresult = dbs.SortedSetScan("key", "an*", pagesize, 0, 0, CommandFlags.None);
}
here I am getting all values of matching criteria instead of page size and next cursor.
so help out if any one has done it before
A:
This is because of stack stackexchange.redis library code. its scanning as per enumerable method. so its not working same as redis command line.
To solve this issue we have used another redis client library called csredis
using (var redis = new RedisClient("yourhost"))
{
string ping = redis.Ping();
var scanresult=redis.ZScan(key, cursor, pattern, pagesize);
}
As shown in above code we will get all dadta into "scanresult".
|
Soluble di- and aminopeptidases in Escherichia K-12. Dispensible enzymes.
As part of a study of the peptidase content of Escherichia coli K-12, two peptidase-deficient amino acid auxotrophs isolated and characterized by Miller as pepD- (strain CM17) and pepD- pepN- pepA- pepB- pepQ- (strain CM89) were examined for the presence of several peptidases previously obtained from strain K-12 in this laboratory. The soluble fraction of each mutant was found to lack the broad-specificity strain K-12 dipeptidase DP and the strain CM89 fraction also lacked activity characteristic of the strain K-12 aminopeptidases AP, L, and OP; like strain CM17, strain CM89 contained the tripeptide-specific aminopeptidase TP. Strain CM89 (but not CM17) appeared to contain little if any activity attributable to the ribosome-bound aminopeptidase I of strain K-12. Whereas loss of DP, AP, OP, and aminopeptidase I activity may be attributed to the pepD-, pepB-, pepN-, and pepA- mutations, respectively, the reason for the loss of L activity remains uncertain. Grown responses of strain CM89 in liquid media containing di- or tripeptides were in accord with absence of enzymes catalyzing rapid hydrolysis of dipeptides. In synthetic liquid media supplemented with the required amino acids per se or with peptone, cultures of both CM strains grew more slowly than strain K-12 and produced smaller cell-yields than those produced by strain K-12. |
1. Field of the Invention
The present invention relates generally to wireless communication systems, and more particularly, to the reporting of Power Headroom (PH) from a User Equipment (UE) in a wireless communication system that supports carrier aggregation.
2. Description of the Related Art
Mobile communication systems were originally designed to provide users with voice communication services while they are on the move. Current mobile communication systems are capable of supporting both voice communication services and data communication services for mobile users.
Standardization for a next generation of mobile communication technology for the 3rd Generation Partnership Project (3GPP) is being conducted for Long Term Evolution (LTE). LTE is a broadband packet-based communication technology that is expected to provide download speeds that improve upon existing data transmission rates by up to 100 Megabytes/second (Mbps). In attempting to achieve such a high data rate, studies have been conducted that use a minimum number of nodes in connection with a simplified network topology, and that place a radio protocol as close as possible to radio channels.
FIG. 1 is a diagram illustrating an LTE wireless communication system. The LTE wireless communication system includes a plurality of Evolved Node Bs (ENBs) 105, 110, 115 and 120, a Mobility Management Entity (MME) 125, and a Serving Gateway (S-GW) 130. ENBs 105, 110, 115 and 120 are coupled to the S-GW 130, enabling a UE 135 to connect to a core network. The ENBs 105, 110, 115 and 120 correspond to Node Bs of a Universal Mobile Telecommunications System (UMTS) and perform more complex functions than those of a legacy Node B. In the LTE system, all user traffic, including real time services such as Voice over Internet Protocol (VoIP), are provided through a shared channel. Each of the ENBs 105, 110, 115 and 120 manage one or more cells, and are responsible for the collection of status information from UEs and for the scheduling of traffic.
In order to support transmission bandwidths of up to 20 megahertz (MHz), LTE employs Orthogonal Frequency Division Multiplexing (OFDM) as its basic modulation scheme. LTE also uses Adaptive Modulation and Coding (AMC) to improve data throughput. AMC varies downlink modulation and coding schemes based on channel conditions for each UE. The S-GW 130 is responsible for managing data bearers and establishes or releases data bearers under the control of the MME 125. The MME 125 is in communication with the S-GW 130 and is responsible for control plane functions.
FIG. 2 is a diagram illustrating a user plane protocol stack for use in the LTE architecture of FIG. 1. A mobile terminal, or UE, 200 has a protocol stack having a Packet Data Convergence Protocol (PDCP) layer 205, a Radio Link Control (RLC) layer 210, a Media Access Control (MAC) layer 215, and a Physical (PHY) layer 220. A base station, or ENB, 201 has a protocol stack having a PDCP layer 240, an RLC layer 235, a MAC layer 230, and a PHY layer 225. The PDCP layers 205 and 240 are responsible for Internet Protocol (IP) header compression/decompression. The RLC layers 210 and 235 pack the PDCP Packet Data Units (PDUs) into a size appropriate for transmission and perform an Automatic Repeat reQuest (ARQ) function. The MAC layers 215 and 230 serve multiple RLC layer entities. These layers are capable of multiplexing the RLC PDUs into a MAC PDU, and demultiplexing the MAC PDU into the RLC PDUs. The PHY layers 220 and 225 perform encoding and modulation on upper layer data for transmission through a radio channel, and perform demodulation and decoding on the OFDM symbol received through the radio channel for delivery to upper layers. A data unit that is input to a protocol entity is referred to as a Service Data Unit (SDU) and a data unit that is output from the protocol entity is referred to as a Protocol Data Unit.
A voice communication service of a wireless communication system requires a relatively small amount of dedicated bandwidth. However, a data communication service must allocate resources in consideration of a data amount and a channel condition so that transmission throughput may increase. Thus, a mobile communication system is provided with a scheduler that manages resource allocation with respect to available resources, channel conditions, an amount of transmission data, etc. Resource scheduling is also required in LTE, and a scheduler that is incorporated into a base station, or ENB, is used to manage radio transmission resources.
In order to meet International Mobile Telephony (IMT)-Advanced requirements that extend beyond those of IMT-2000, further technological advancements have allowed for the evolution of LTE into LTE-Advanced (LTE-A). LTE-A is provided with technological components, such as carrier aggregation, to fulfill the IMT-Advanced requirements. Carrier aggregation aggregates multiple carriers to form a larger bandwidth, thereby allowing a UE to transmit and receive data at higher data rates.
FIG. 3 is a schematic diagram illustrating an LTE-A wireless communication system supporting carrier aggregation. An ENB 305 operates on two different carriers 310 and 315, having center frequencies of f3 and f1, respectively. A conventional wireless communication system allows a UE 330 to communicate with the ENB 305 using only one of carriers 310 and 315. However, the LTE-A system supporting carrier aggregation enables the UE 330 to use both carriers 310 and 315 in order to increase transmission throughput. The maximum data rate between the ENB 305 and the UE 330 increases in proportion to the number of carriers that are aggregated.
Due to the fact that uplink transmissions cause inter-cell interference, it is preferable for a UE to calculate an uplink transmission power using a predetermined function, and to control uplink transmission based on the calculation. The predetermined function may utilize variables such as an allocated transmission resource amount, a Modulation and Coding Scheme (MCS), and a path loss value in calculating a required uplink transmission power. The uplink transmission power is limited to a UE maximum transmission power. When the required uplink transmission power is greater than the UE maximum transmission power, the UE performs the uplink transmission using the UE maximum transmission power. However, use of the maximum transmission power instead of the required transmission power degrades the uplink transmission quality. Thus, it is preferable for the ENB to perform scheduling for UE transmissions such that a required transmission power for the UE transmission will not exceed the UE maximum transmission power.
Some parameters utilized in scheduling at the ENB, such as channel path loss, are not capable of being measured at the ENB. When required, the UE may transmit a Power Headroom Report (PHR) to the ENB to report UE Power Headroom (PH) with respect to path loss. However, conventional uplink transmission power determination procedures are performed with respect to a single downlink carrier and a single uplink carrier. Thus, the conventional procedures are not applicable to the LTE-A system supporting carrier aggregation. |
# Project-wide Gradle settings.
# IDE (e.g. Android Studio) users:
# Gradle settings configured through the IDE *will override*
# any settings specified in this file.
# For more details on how to configure your build environment visit
# http://www.gradle.org/docs/current/userguide/build_environment.html
# Specifies the JVM arguments used for the daemon process.
# The setting is particularly useful for tweaking memory settings.
org.gradle.jvmargs=-Xmx1024m
# When configured, Gradle will run in incubating parallel mode.
# This option should only be used with decoupled projects. More details, visit
# http://www.gradle.org/docs/current/userguide/multi_project_builds.html#sec:decoupled_projects
# org.gradle.parallel=true
|
Neighbors (novel)
Neighbors is a 1980 novel by American author Thomas Berger. It is a satire of manners and suburbia, and a comment on emotional alienation with echoes of the works of Franz Kafka. Earl Keese’s character and situation begin realistically but become increasingly fantastic. Keese is an Everyman whose life is swiftly turned upside down. As he scrambles to reclaim his sense of normalcy and dignity, he comes to think that everyone, including his family, is against him.
Plot summary
Earl Keese is a middle-aged, middle-class suburbanite with a wife, Enid, and teenage daughter, Elaine. Earl is content with his dull, unexceptional life, but this changes when a younger, less sophisticated couple, Harry and Ramona, move in next door. Harry is physically intimidating and vulgar; Ramona is sexually aggressive, and both impose themselves on the Keese household. Their free-spirited personalities and overbearing and boorish behavior endear them to Enid and Elaine, but Earl fears that he is losing control of his life and his family. Over the course of one night, the antagonism between Earl and his new neighbors escalates into suburban warfare.
Analysis
Berger's off-kilter tone blurs the line between paranoia and reality, defense and offense, action and intention, ally and adversary. Harry and Ramona seem to constantly undergo changes in their respective personalities and Enid and Elaine appear to choose sides against Earl at random, but Berger also implies that it is Earl’s sense of reality that is skewed and deluded.
Earl is frustrated because he can never prove that Harry and Ramona are doing anything wrong on purpose, and the more he attempts to expose them, the more ridiculous he makes himself. Yet Earl comes to realize that Harry and Ramona have served as the crucible of his redemption: being forced out of his comfort zone of complacency and habit has provided him with an excitement he has never known before. As Earl comes to recognize value in his neighbors, he realizes that his wife is a distrustful alcoholic, his daughter is an underachiever and petty thief, and that his new neighbors can provide him with an escape from his existence of insignificance and emotional impotence. From a nightmare comes hope and a strengthened resolve to survive. In his study of Berger, writer Stanley Trachtenberg describes Neighbors as an existentialist parable in which "the loss of coherence between various aspects of self comically fragments the notion of identity and thus fictionalizes the existential concept of authenticity as a shaping condition of it."
In a 1980 newspaper interview, Berger said of Neighbors, "As my 10th novel, begun at the close of my 20th year as a published novelist, it is appropriately a bizarre celebration of whatever gift I have, the strangest of all my narratives . . . the morality of this work, like that of all my other volumes, will be in doubt until the end of the narrative – and perhaps to the end of eternity, now that I think about it."
Characters
Earl Keese
Enid Keese
Elaine Keese
Harry
Ramona
Adaptations
A film version was released in 1981, starring John Belushi and Dan Aykroyd. It was also adapted into a play by Eve Summer, which premiered in Worcester, Massachusetts in 2007.
References
External links
NPR.org | Tom Perrotta Hails Suburban Sendup 'Neighbors'
Category:1980 American novels
Category:American novels adapted into films
Category:American novels adapted into plays
Category:Novels by Thomas Berger (novelist) |
Longitudinal impedance variability in patients with chronically implanted DBS devices.
Deep brain stimulation (DBS) is an effective therapy for advanced movement disorders, but its optimal use is still controversial. One factor that could play a role in the proper delivery of therapeutic stimulation by current DBS devices is the variability of the impedance at the interface between the electrode surface and surrounding tissue. To analyze variability and trends in the impedance of chronically-implanted DBS electrodes in subjects with movement disorders. We reviewed impedance values from medical records of DBS patients at an academic tertiary-care movement disorders center. The standard deviation of data recorded within individual subjects and single contacts were used as measures of longitudinal impedance variability. A generalized linear mixed model (GLMM) determined if a number of effects had significant influences on impedance. We analyzed 2863 impedance measurements from 94 subjects. Median variability, for subjects with follow-up from 6 months to 5 years (n = 77), was 194 Ω for individual subjects and 141 Ω for individual contacts, with a range spanning from 18 to over 600 Ω. The GLMM, incorporating all subjects (n = 94), identified time, electrical activity, implanted target, contact position on the electrode and side of implantation as significant predictors of impedance. Age and disease duration at surgery, gender or ethnicity were not significant predictors. Our analysis suggests that a significant amount of impedance variability can be expected in chronically implanted DBS electrodes and indicates a number of factors with possible predictive value. Further studies are needed to link impedance characteristics to clinical outcomes. |
Music from McLeod's Daughters
McLeod's Daughters have had many different songs for their closing credits which are written by Posie Graeme-Evans & Chris Harriot and performed by singer Rebecca Lavelle who also had a guest role in series 6 as Bindi Martin
Song List
Other
Hey You by Abi Tucker who plays Grace McLeod from 2007 - 2008 and featured the song in Episode 196, My Enemy, My Friend.
List of Released Songs
Rebecca Lavelle
Understand Me
Common Ground
Never Enough
Don't Judge
Love You, Hate You
Heat
Am I Crazy?
We Got It Wrong
The Siren's Song
Hopeless Case
Just A Child
My Heart Is Like A River
Theme Song - Version 1
Hey Girl (You Got A New Life)
Take The Rain Away
The Stranger
Sometimes
Too Young
The First Touch
In His Eyes
By My Side
Did I Tell You?
Don't Give Up
Gentle Gentle (Life of Your Life)
Theme Song - Version 2
You Believed
Had To Happen
It Comes To This
Charlotte's Song
One True Thing
I Wish The Past Was Different
Locked Away Inside My Heart
Our Home, Our Place
Strip Jack Naked
Broken Dreams
This Perfect Day
Trust The Night
The Man I Loved (We Had No Time)
Time Turn Over
Drover's Run (My Heart's Home)
Abi Tucker
Hey You
Speak My Angel
List of Unreleased Songs
Feet on The Ground by Rebecca Lavelle
Room To Move by Rebecca Lavelle
A Matter of Time by Rebecca Lavelle
All I Ever Wanted was Love by Rebecca Lavelle
Alone & Afraid by Rebecca Lavelle
Belonging by Rebecca Lavelle
I Reach Out by Naomi Starr
Life Makes A Fool of Us by Rebecca Lavelle
Love is Endless by Rebecca Lavelle
Something So Strong by Rebecca Lavelle
Sorrow by Rebecca Lavelle
Stay by Rebecca Lavelle
Tears on My Pillow by Rebecca Lavelle & Glenda Linscott
Kate's Lullaby by Michala Banas
Wake Up Gungellan by Doris Younane (Abi Tucker & Gillian Alexy Short Clip)
Truckstop Woman by Doris Younane, Simmone Jade Mackinnon, Luke Jacobz, Gillian Alexy & Chorus
Forever by Doris Youanne, Peter Hardy, Abi Tucker & Matt Passmore
References
External links
McLeod's Daughters Official Website
Dutch McLeod's Daughters Website
Category:McLeod's Daughters |
Q:
How to create a django User using DRF's ModelSerializer
In django, creating a User has a different and unique flow from the usual Model instance creation. You need to call create_user() which is a method of BaseUserManager.
Since django REST framework's flow is to do restore_object() and then save_object(), it's not possible to simply create Users using a ModelSerializer in a generic create API endpoint, without hacking you way through.
What would be a clean way to solve this? or at least get it working using django's built-in piping?
Edit:
Important to note that what's specifically not working is that once you try to authenticate the created user instance using django.contrib.auth.authenticate it fails if the instance was simply created using User.objects.create() and not .create_user().
A:
Eventually I've overridden the serializer's restore_object method and made sure that the password being sent is then processes using instance.set_password(password), like so:
def restore_object(self, attrs, instance=None):
if not instance:
instance = super(RegisterationSerializer, self).restore_object(attrs, instance)
instance.set_password(attrs.get('password'))
return instance
Thanks everyone for help!
|
The long term objective is to characterize key functionalities of the epithelial cells of the larval mosquito gut as these cellular functions influence and regulate the anionic basis of alkalinization of the gut lumen. A detailed understanding of how gut epithelial cells produce the remarkable and biologically unique pH extremes (i.e. > 10.5) that drive the digestive process will provide new avenues for the development of environmentally safe and specific larvacides. Two specific gene families have been targeted as they have central roles in anion production and transport in the gut alkalinization process: carbonic anhydrases and transmembrane anion transporters. This project will produce molecular and physiological characterizations of members of these two gene families. Their distributions in the tissue and specific roles in larval mosquito gut alkalinization will be defined. Specific cellular phenotypes throughout the gut will be defined and the role of each in the alkalinization process assessed. AIM 1 will examine the expression of multiple carbonic anhydrases. AIM 2 will define and characterize members of the anion transporter gene family. AIM 3 will define the cellular distributions of carbonic anhydrases and anion transporters in the gut and as functions of larval development. AIM 4 will produce a global analysis of gene expression in the specific functional domains of the larval mosquito gut identifying key functionalities which define the gut domains. AIM 5 will bring the localization of specific gene products together with physiological measurements of the activity of individual cells to produce a cell-specific and spatial analysis of anion dynamics in the gut epithelium. As mosquitoes are the number one threat to human health world wide and recognized as potential agents for bioterrorism, the development of new strategies for control based on unique aspects of their biology (i.e. gut alkalinization) has important potential. [unreadable] [unreadable] [unreadable] |
Effect of two prophylaxis methods on adherence of Streptococcus mutans to microfilled composite resin and giomer surfaces.
Surface attributes of a restoration play an important role in adherence of plaque bacteria. Prophylaxis methods may be involved in modification of or damaging the restoration surface. The aim of the present study was to evaluate the effect of two prophylaxis methods on adherence of Streptococcus mutans to the surface of two restorative materials. A total of 60 specimens were prepared from each material; a microfilled composite resin (HelioProgress) and a giomer (Beautifil II). For each material, the specimens were randomly divided into three groups (n=20). Group 1: no prophylaxis treatment (control); Group 2: prophylaxis with pumice and rubber cup; Group 3: prophylaxis with air-powder polishing device (APD). The surfaces of selected specimens from each group were evaluated under a scanning electron microscope (SEM), and the surface topography formed by the two prophylaxis methods was determined by atomic force microscopy (AFM). Adherence of Streptococcus mutans to the surface of specimens was determined by the plate counting method following immersion in a bacterial innoculum for 4 hours, rinsing and sonication. Data were analyzed by two-way ANOVA and post hoc Tukey test for multiple comparisons. Statistical significance was set at P<0.05. Bacterial adherence was significantly affected by both factors: restorative material type and prophylaxis method (P<0.0005). Mean bacterial adhesion was significantly higher in composite groups compared to corresponding giomer groups. Within each material, bacterial adherence was significantly lower in the control group compared to prophylaxis groups. Prophylaxis with pumice and rubber cup resulted in a significantly lower bacterial adherence compared to prophylaxis with APD. Based on the results of the present study, giomer specimens demonstrated lower bacterial adherence compared to composite resin specimens. In both materials, the highest bacterial adherence was observed with prophylaxis with APD, pumice and rubber cup and the control group, respectively. |
---
abstract: |
We give a general construction of debiased/locally robust/orthogonal (LR) moment functions for GMM, where the derivative with respect to first step nonparametric estimation is zero and equivalently first step estimation has no effect on the influence function. This construction consists of adding an estimator of the influence function adjustment term for first step nonparametric estimation to identifying or original moment conditions. We also give numerical methods for estimating LR moment functions that do not require an explicit formula for the adjustment term.
LR moment conditions have reduced bias and so are important when the first step is machine learning. We derive LR moment conditions for dynamic discrete choice based on first step machine learning estimators of conditional choice probabilities.
We provide simple and general asymptotic theory for LR estimators based on sample splitting. This theory uses the additive decomposition of LR moment conditions into an identifying condition and a first step influence adjustment. Our conditions require only mean square consistency and a few (generally either one or two) readily interpretable rate conditions.
LR moment functions have the advantage of being less sensitive to first step estimation. Some LR moment functions are also doubly robust meaning they hold if one first step is incorrect. We give novel classes of doubly robust moment functions and characterize double robustness. For doubly robust estimators our asymptotic theory only requires one rate condition.
Keywords: Local robustness, orthogonal moments, double robustness, semiparametric estimation, bias, GMM.
JEL classification:
: C13; C14; C21; D24
author:
- |
Victor Chernozhukov\
*MIT*
- |
Juan Carlos Escanciano\
*Indiana University*
- |
Hidehiko Ichimura\
*University of Tokyo*
- |
Whitney K. Newey\
*MIT*
- |
James M. Robins\
*Harvard University*
date: April 2018
title: Locally Robust Semiparametric Estimation
---
Introduction
============
There are many economic parameters that depend on nonparametric or large dimensional first steps. Examples include dynamic discrete choice, games, average consumer surplus, and treatment effects. This paper shows how to construct moment functions for GMM estimators that are debiased/locally robust/orthogonal (LR), where moment conditions have a zero derivative with respect to the first step. We show that LR moment functions can be constructed by adding the influence function adjustment for first step estimation to the original moment functions. This construction can also be interpreted as a decomposition of LR moment functions into identifying moment functions and a first step influence function term. We use this decomposition to give simple and general conditions for root-n consistency and asymptotic normality, with different properties being assumed for the identifying and influence function terms. The conditions are easily interpretable mean square consistency and second order remainder conditions based on estimated moments that use cross-fitting (sample splitting). We also give numerical estimators of the influence function adjustment.
LR moment functions have several advantages. LR moment conditions bias correct in a way that eliminates the large biases from plugging in first step machine learning estimators found in Belloni, Chernozhukov, and Hansen (2014). LR moment functions can be used to construct debiased/double machine learning (DML) estimators, as in Chernozhukov et al. (2017, 2018).
We illustrate by deriving LR moment functions for dynamic discrete choice estimation based on conditional choice probabilities. We provide a DML estimator for dynamic discrete choice that uses first step machine learning of conditional choice probabilities. We find that it performs well in a Monte Carlo example. Such structural models provide a potentially important application of DML, because of potentially high dimensional state spaces. Adding the first step influence adjustment term provides a general way to construct LR moment conditions for structural models so that machine learning can be used for first step estimation of conditional choice probabilities, state transition distributions, and other unknown functions on which structural estimators depend.
LR moment conditions also have the advantage of being relatively insensitive to small variation away from the first step true function. This robustness property is appealing in many settings where it may be difficult to get the first step completely correct. Many interesting and useful LR moment functions have the additional property that they are doubly robust (DR), meaning moment conditions hold when one first step is not correct. We give novel classes of DR moment conditions, including for average linear functionals of conditional expectations and probability densities. The construction of adding the first step influence function adjustment to an identifying moment function is useful to obtain these moment conditions. We also give necessary and sufficient conditions for a large class of moment functions to be DR. We find DR moments have simpler and more general conditions for asymptotic normality, which helps motivate our consideration of DR moment functions as special cases of LR ones. LR moment conditions also help minimize sensitivity to misspecification as in Bonhomme and Weidner (2018).
LR moment conditions have smaller bias from first step estimation. We show that they have the small bias property of Newey, Hsieh, and Robins (2004), that the bias of the moments is of smaller order than the bias of the first step. This bias reduction leads to substantial improvements in finite sample properties in many cases relative to just using the original moment conditions. For dynamic discrete choice we find large bias reductions, moderate variance increases and even reductions in some cases, and coverage probabilities substantially closer to nominal. For machine learning estimators of the partially linear model, Chernozhukov et al. (2017, 2018) found bias reductions so large that the LR estimator is root-n consistent but the estimator based on the original moment condition is not. Substantial improvements were previously also found for density weighted averages by Newey, Hsieh, and Robins (2004, NHR). The twicing kernel estimators in NHR are numerically equal to LR estimators based on the original (before twicing) kernel, as shown in Newey, Hsieh, Robins (1998), and the twicing kernel estimators were shown to have smaller mean square error in large samples. Also, a Monte Carlo example in NHR finds that the mean square error (MSE) of the LR estimator has a smaller minimum and is flatter as a function of bandwidth than the MSE of Powell, Stock, and Stoker’s (1989) density weighted average derivative estimator. We expect similar finite sample improvements from LR moments in other cases.
LR moment conditions have appeared in earlier work. They are semiparametric versions of Neyman (1959) C-alpha test scores for parametric models. Hasminskii and Ibragimov (1978) suggested LR estimation of functionals of a density and argued for their advantages over plug-in estimators. Pfanzagl and Wefelmeyer (1981) considered using LR moment conditions for improving the asymptotic efficiency of functionals of distribution estimators. Bickel and Ritov (1988) gave a LR estimator of the integrated squared density that attains root-n consistency under minimal conditions. The Robinson (1988) semiparametric regression and Ichimura (1993) index regression estimators are LR. Newey (1990) showed that LR moment conditions can be obtained as residuals from projections on the tangent set in a semiparametric model. Newey (1994a) showed that derivatives of an objective function where the first step has been “concentrated out” are LR, including the efficient score of a semiparametric model. NHR (1998, 2004) gave estimators of averages that are linear in density derivative functionals with remainder rates that are as fast as those in Bickel and Ritov (1988). Doubly robust moment functions have been constructed by Robins, Rotnitzky, and Zhao (1994, 1995), Robins and Rotnitzky (1995), Scharfstein, Rotnitzky, and Robins (1999), Robins, Rotnitzky, and van der Laan (2000), Robins and Rotnitzky (2001), Graham (2011), and Firpo and Rothe (2017). They are widely used for estimating treatment effects, e.g. Bang and Robins (2005). Van der Laan and Rubin (2006) developed targeted maximum likelihood to obtain a LR estimating equation based on the efficient influence function of a semiparametric model. Robins et al. (2008, 2017) showed that efficient influence functions are LR, characterized some doubly robust moment conditions, and developed higher order influence functions that can reduce bias. Belloni, Chernozhukov, and Wei (2013), Belloni, Chernozhukov, and Hansen (2014), Farrell (2015), Kandasamy et al. (2015), Belloni, Chernozhukov, Fernandez-Val, and Hansen (2016), and Athey, Imbens, and Wager (2017) gave LR estimators with machine learning first steps in several specific contexts.
A main contribution of this paper is the construction of LR moment conditions from any moment condition and first step estimator that can result in a root-n consistent estimator of the parameter of interest. This construction is based on the limit of the first step when a data observation has a general distribution that allows for misspecification, similarly to Newey (1994). LR moment functions are constructed by adding to identifying moment functions the influence function of the true expectation of the identifying moment functions evaluated at the first step limit, i.e. by adding the influence function term that accounts for first step estimation. The addition of the influence adjustment “partials out” the first order effect of the first step on the moments. This construction of LR moments extends those cited above for first step density and distribution estimators to *any first step,* including instrumental variable estimators. Also, this construction is *estimator based* rather than model based as in van der Laan and Rubin (2006) and Robins et al. (2008, 2017). The construction depends only on the moment functions and the first step rather than on a semiparametric model. Also, we use the fundamental Gateaux derivative definition of the influence function to show LR rather than an embedding in a regular semiparametric model.
The focus on the functional that is the true expected moments evaluated at the first step limit is the key to this construction. This focus should prove useful for constructing LR moments in many setting, including those where it has already been used to find the asymptotic variance of semiparametric estimators, such as Newey (1994a), Pakes and Olley (1995), Hahn (1998), Ai and Chen (2003), Hirano, Imbens, and Ridder (2003), Bajari, Hong, Krainer, and Nekipelov (2010), Bajari, Chernozhukov, Hong, and Nekipelov (2009), Hahn and Ridder (2013, 2016), and Ackerberg, Chen, Hahn, and Liao (2014), Hahn, Liao, and Ridder (2016). One can construct LR moment functions in each of these settings by adding the first step influence function derived for each case as an adjustment to the original, identifying moment functions.
Another contribution is the development of LR moment conditions for dynamic discrete choice. We derive the influence adjustment for first step estimation of conditional choice probabilities as in Hotz and Miller (1993). We find encouraging Monte Carlo results when various machine learning methods are used to construct the first step. We also give LR moment functions for conditional moment restrictions based on orthogonal instruments.
An additional contribution is to provide general estimators of the influence adjustment term that can be used to construct LR moments without knowing their form. These methods estimate the adjustment term numerically, thus avoiding the need to know its form. It is beyond the scope of this paper to develop machine learning versions of these numerical estimators. Such estimators are developed by Chernozhukov, Newey, and Robins (2018) for average linear functionals of conditional expectations.
Further contributions include novel classes of DR estimators, including linear functionals of nonparametric instrumental variables and density estimators, and a characterization of (necessary and sufficient conditions for) double robustness. We also give related, novel partial robustness results where original moment conditions are satisfied even when the first step is not equal to the truth.
A main contribution is simple and general asymptotic theory for LR estimators that use cross-fitting in the construction of the average moments. This theory is based on the structure of LR moment conditions as an identifying moment condition depending on one first step plus an influence adjustment that can depend on an additional first step. We give a remainder decomposition that leads to mean square consistency conditions for first steps plus a few readily interpretable rate conditions. For DR estimators there is only one rate condition, on a product of sample remainders from two first step estimators, leading to particularly simple conditions. This simplicity motivates our inclusion of results for DR estimators. This asymptotic theory is also useful for existing moment conditions that are already known to be LR. Whenever the moment condition can be decomposed into an identifying moment condition depending on one first step and an influence function term that may depend on two first steps the simple and general regularity conditions developed here will apply.
LR moments reduce that smoothing bias that results from first step nonparametric estimation relative to original moment conditions. There are other sources of bias arising from nonlinearity of moment conditions in the first step and the empirical distribution. Cattaneo and Jansson (2017) and Cattaneo, Jansson, and Ma (2017) give useful bootstrap and jackknife methods that reduce nonlinearity bias. Newey and Robins (2017) show that one can also remove this bias by cross fitting in some settings. We allow for cross-fitting in this paper.
Section 2 describes the general construction of LR moment functions for semiparametric GMM. Section 3 gives LR moment conditions for dynamic discrete choice. Section 4 shows how to estimate the first step influence adjustment. Section 5 gives novel classes of DR moment functions and characterizes double robustness. Section 6 gives an orthogonal instrument construction of LR moments based on conditional moment restrictions. Section 7 provides simple and general asymptotic theory for LR estimators.
Locally Robust Moment Functions
===============================
The subject of this paper is GMM estimators of parameters where the sample moment functions depend on a first step nonparametric or large dimensional estimator. We refer to these estimators as semiparametric. We could also refer to them as GMM where first step estimators are plugged in the moments. This terminology seems awkward though, so we simply refer to them as semiparametric GMM estimators. We denote such an estimator by $\hat{\beta}$, which is a function of the data $z_{1},...,z_{n}$ where $n$ is the number of observations. Throughout the paper we will assume that the data observations $z_{i}$ are i.i.d. We denote the object that $\hat{\beta}$ estimates as $\beta_{0}$, the subscript referring to the parameter value under the distribution $F_{0}$ of $z_{i}$.
To describe semiparametric GMM let $m(z,\beta,\gamma)$ denote an $r\times1$ vector of functions of the data observation $z,$ parameters of interest $\beta$, and a function $\gamma$ that may be vector valued. The function $\gamma$ can depend on $\beta$ and $z$ through those arguments of $m.$ Here the function $\gamma$ represents some possible first step, such as an estimator, its limit, or a true function. A GMM estimator can be based on a moment condition where $\beta_{0}$ is the unique parameter vector satisfying$$E[m(z_{i},\beta_{0},\gamma_{0})]=0, \label{moments}$$ and $\gamma_{0}$ is the true $\gamma$. We assume that this moment condition identifies $\beta.$ Let $\hat{\gamma}$ denote some first step estimator of $\gamma_{0}$. Plugging in $\hat{\gamma}$ to obtain $m(z_{i},\beta,\hat{\gamma
})$ and averaging over $z_{i}$ results in the estimated sample moments $\hat{m}(\beta)=\sum_{i=1}^{n}m(z_{i},\beta,\hat{\gamma})/n.$ For $\hat{W}$ a positive semi-definite weighting matrix a semiparametric GMM estimator is$$\tilde{\beta}=\arg\min_{\beta\in B}\hat{m}(\beta)^{T}\hat{W}\hat{m}(\beta),$$ where $A^{T}$ denotes the transpose of a matrix $A$ and $B$ is the parameter space for $\beta$. Such estimators have been considered by, e.g. Andrews (1994), Newey (1994a), Newey and McFadden (1994), Pakes and Olley (1995), Chen and Liao (2015), and others.
Locally robust (LR) moment functions can be constructed by adding the influence function adjustment for the first step estimator $\hat{\gamma}$ to the identifying or original moment functions $m(z,\beta,\gamma).$ To describe this influence adjustment let $\gamma(F)$ denote the limit of $\hat{\gamma}$ when $z_{i}$ has distribution $F,$ where we restrict $F$ only in that $\gamma(F)$ exists and possibly other regularity conditions are satisfied. That is, $\gamma(F)$ is the limit of $\hat{\gamma}$ under possible misspecification, similar to Newey (1994). Let $G$ be some other distribution and $F_{\tau}=(1-\tau)F_{0}+\tau G$ for $0\leq\tau\leq1,$ where $F_{0}$ denotes the true distribution of $z_{i}.$ We assume that $G$ is chosen so that $\gamma(F_{\tau})$ is well defined for $\tau>0$ small enough and possibly other regularity conditions are satisfied, similarly to Ichimura and Newey (2017). The influence function adjustment will be the function $\phi
(z,\beta,\gamma,\lambda)$ such that for all such $G,$$$\frac{d}{d\tau}E[m(z_{i},\beta,\gamma(F_{\tau}))]=\int\phi(z,\beta,\gamma
_{0},\lambda_{0})G(dz),E[\phi(z_{i},\beta,\gamma_{0},\lambda_{0})]=0,
\label{infdef}$$ where $\lambda$ is an additional nonparametric or large dimensional unknown object on which $\phi(z,\beta,\gamma,\lambda)$ depends and the derivative is from the right (i.e. for positive values of $\tau$) and at $\tau=0.$ This equation is the well known definition of the influence function $\phi
(z,\beta,\gamma_{0},\lambda_{0})$ of $\mu(F)=E[m(z_{i},\beta,\gamma(F))]$ as the Gateaux derivative of $\mu(F),$ e.g. Huber (1981). The restriction of $G$ so that $\gamma(F_{\tau})$ exists allows $\phi(z,\beta,\gamma_{0},\lambda
_{0})$ to be the influence function when $\gamma(F)$ is only well defined for certain types of distributions, such as when $\gamma(F)$ is a conditional expectation or density. The function $\phi(z,\beta,\gamma,\lambda)$ will generally exist when $E[m(z_{i},\beta,\gamma(F))]$ has a finite semiparametric variance bound. Also $\phi(z,\beta,\gamma,\lambda)$ will generally be unique because we are not restricting $G$ very much. Also, note that $\phi
(z,\beta,\gamma,\lambda)$ will be the influence adjustment term from Newey (1994a), as discussed in Ichimura and Newey (2017).
LR moment functions can be constructed by adding $\phi(z,\beta,\gamma
,\lambda)$ to $m(z,\beta,\gamma)$ to obtain new moment functions$$\psi(z,\beta,\gamma,\lambda)=m(z,\beta,\gamma)+\phi(z,\beta,\gamma,\lambda).
\label{momadj}$$ Let $\hat{\lambda}$ be a nonparametric or large dimensional estimator having limit $\lambda(F)$ when $z_{i}$ has distribution $F,$ with $\lambda
(F_{0})=\lambda_{0}.$ Also let $\hat{\psi}(\beta)=\sum_{i=1}^{n}\psi
(z_{i},\beta,\hat{\gamma},\hat{\lambda})/n.$ A LR GMM estimator can be obtained as$$\hat{\beta}=\arg\min_{\beta\in B}\hat{\psi}(\beta)^{T}\hat{W}\hat{\psi}(\beta). \label{lrgmm}$$ As usual a choice of $\hat{W}$ that minimizes the asymptotic variance of $\sqrt{n}(\hat{\beta}-\beta_{0})$ will be a consistent estimator of the inverse of the asymptotic variance $\Omega$ of $\sqrt{n}\hat{\psi}(\beta
_{0}).$ As we will further discuss, $\psi(z,\beta,\gamma,\lambda)$ being LR will mean that the estimation of $\gamma$ and $\lambda$ does not affect $\Omega$, so that $\Omega=E[\psi(z_{i},\beta_{0},\gamma_{0},\lambda_{0})\psi(z_{i},\beta_{0},\gamma_{0},\lambda_{0})^{T}].$ An optimal $\hat{W}$ also gives an efficient estimator in the wider sense shown in Ackerberg, Chen, Hahn, and Liao (2014), making $\hat{\beta}$ efficient in a semiparametric model where the only restrictions imposed are equation (\[moments\]).
The LR property we consider is that the derivative of the true expectation of the moment function with respect to the first step is zero, for a Gateaux derivative like that for the influence function in equation (\[infdef\]). Define $F_{\tau}=(1-\tau)F_{0}+\tau G$ as before where $G$ is such that both $\gamma(F_{\tau})$ and $\lambda(F_{\tau})$ are well defined. The LR property is that for all $G$ as specified,$$\frac{d}{d\tau}E[\psi(z_{i},\beta,\gamma(F_{\tau}),\lambda(F_{\tau}))]=0.
\label{lrdef}$$ Note that this condition is the same as that of Newey (1994a) for the presence of $\hat{\gamma}$ an $\hat{\lambda}$ to have no effect on the asymptotic distribution, when each $F_{\tau}$ is a regular parametric submodel. Consequently, the asymptotic variance of $\sqrt{n}\hat{\psi}(\beta_{0})$ will be $\Omega$ as in the last paragraph.
To show LR of the moment functions $\psi(z,\beta,\gamma,\lambda)=m(z,\beta
,\gamma)+\phi(z,\beta,\gamma,\lambda)$ from equation (\[momadj\]) we use the fact that the second, zero expectation condition in equation (\[infdef\]) must hold for all possible true distributions. For any given $\beta$ define $\mu(F)=E[m(z_{i},\beta,\gamma(F))]$ and $\phi(z,F)=\phi(z,\beta
,\gamma(F),\lambda(F)).$
<span style="font-variant:small-caps;">Theorem 1:</span> *If i)* $d\mu(F_{\tau})/d\tau=\int\phi
(z,F_{0})G(dz)$*, ii)* $\int\phi(z,F_{\tau})F_{\tau}(dz)=0$ *for all* $\tau\in\lbrack0,\bar{\tau}),$ *and iii)* $\int\phi(z,F_{\tau
})F_{0}(dz)$ *and* $\int\phi(z,F_{\tau})G(dz)$ *are continuous at* $\tau=0$ *then*$$\frac{d}{d\tau}E[\phi(z_{i},F_{\tau})]=-\frac{d\mu(F_{\tau})}{d\tau}.
\label{thm1con}$$
The proofs of this result and others are given in Appendix B. Assumptions i) and ii) of Theorem 1 require that both parts of equation (\[infdef\]) hold with the second, zero mean condition being satisfied when $F_{\tau}$ is the true distribution. Assumption iii) is a regularity condition. The LR property follows from Theorem 1 by adding $d\mu(F_{\tau})/d\tau$ to both sides of equation (\[thm1con\]) and noting that the sum of derivatives is the derivative of the sum. Equation (\[thm1con\]) shows that the addition of $\phi(z,\beta,\gamma,\lambda)$ “partials out” the effect of the first step $\gamma$ on the moment by “cancelling” the derivative of the identifying moment $E[m(z_{i},\beta,\gamma(F_{\tau}))]$ with respect to $\tau$. This LR result for $\psi(z,\beta,\gamma,\lambda)$ differs from the literature in its Gateaux derivative formulation and in the fact that it is not a semiparametric influence function but is the hybrid sum of an identifying moment function $m(z,\beta,\gamma)$ and an influence function adjustment $\phi(z,\beta
,\gamma,\lambda).$
Another zero derivative property of LR moment functions is useful. If the sets $\Gamma$ and $\Lambda$ of possible limits $\gamma(F)$ and $\lambda(F)$, respectively, are linear, $\gamma(F)$ and $\lambda(F)$ can vary separately from one another, and certain functional differentiability conditions hold then LR moment functions will have the property that for any $\gamma\in\Gamma
$, $\lambda\in\Lambda$, and $\bar{\psi}(\gamma,\lambda)=E[\psi(z_{i},\beta
_{0},\gamma,\lambda)]$, $$\frac{\partial}{\partial\tau}\bar{\psi}((1-\tau)\gamma_{0}+\tau\gamma
,\lambda_{0})=0,\frac{\partial}{\partial\tau}\bar{\psi}(\gamma_{0},(1-\tau)\lambda_{0}+\tau\lambda)=0. \label{lrdef2}$$ That is, the expected value of the LR moment function will have a zero Gateaux derivative with respect to each of the first steps $\gamma$ and $\lambda.$ This property will be useful for several results to follow. Under still stronger smoothness conditions this zero derivative condition will result in the existence of a constant $C$ such that for a function norm $\left\Vert
\cdot\right\Vert $,$$\left\vert \bar{\psi}(\gamma,\lambda_{0})\right\vert \leq C\left\Vert
\gamma-\gamma_{0}\right\Vert ^{2},\text{ }\left\vert \bar{\psi}(\gamma
_{0},\lambda)\right\vert \leq C\left\Vert \lambda-\lambda_{0}\right\Vert ^{2},
\label{nlremainder}$$ when $\left\Vert \gamma-\gamma_{0}\right\Vert $ and $\left\Vert \lambda
-\lambda_{0}\right\Vert $ are small enough. In Appendix B we give smoothness conditions that are sufficient for LR to imply equations (\[lrdef2\]) and (\[nlremainder\]). When formulating regularity conditions for particular moment functions and first step estimators it may be more convenient to work directly with equation (\[lrdef2\]) and/or (\[nlremainder\]).
The approach of constructing LR moment functions by adding the influence adjustment differs from the model based approach of using an efficient influence function or score for a semiparametric model as moment functions . The approach here is *estimator based* rather than model based. The influence adjustment $\phi(z,\beta,\gamma,\lambda)$ is determined by the limit $\gamma(F)$ of the first step estimator $\hat{\gamma}$ and the moment functions $m(z,\beta,\gamma)$ rather than by some underlying semiparametric model. This estimator based approach has proven useful for deriving the influence function of a wide variety of semiparametric estimators, as mentioned in the Introduction. Here this estimator based approach provides a general way to construct LR moment functions. For any moment function $m(z,\beta,\gamma)$ and first step estimator $\hat{\gamma}$ a corresponding LR estimator can be constructed as in equations (\[momadj\]) and (\[lrgmm\]).
The addition of $\phi(z,\beta,\gamma,\lambda)$ does not affect identification of $\beta$ because $\phi(z,\beta,\gamma_{0},\lambda_{0})$ has expectation zero for any $\beta$ and true $F_{0}.$ Consequently, the LR GMM estimator will have the same asymptotic variance as the original GMM estimator $\tilde{\beta}$ when $\sqrt{n}(\tilde{\beta}-\beta_{0})$ is asymptotically normal, under appropriate regularity conditions. The addition of $\phi(z,\beta
,\gamma,\lambda)$ will change other properties of the estimator. As discussed in Chernozhukov et al. (2017, 2018), it can even remove enough bias so that the LR estimator is root-n consistent and the original estimator is not.
If $F_{\tau}$ was modified so that $\tau$ is a function of a smoothing parameter, e.g. a bandwidth, and $\tau$ gives the magnitude of the smoothing bias of $\gamma(F_{\tau}),$ then equation (\[lrdef\]) is a small bias condition, equivalent to$$E[\psi(z_{i},\beta_{0},\gamma(F_{\tau}),\lambda(F_{\tau}))]=o(\tau).$$ Here $E[\psi(z_{i},\beta_{0},\gamma(F_{\tau}),\lambda(F_{\tau}))]$ is a bias in the moment condition resulting from smoothing that shrinks faster than $\tau.$ In this sense LR GMM estimators have the small bias property considered in NHR. This interpretation is also one sense in which LR GMM is “debiased.”
In some cases the original moment functions $m(z,\beta,\gamma)$ are already LR and the influence adjustment will be zero. An important class of moment functions that are LR are those where $m(z,\beta,\gamma)$ is the derivative with respect to $\beta$ of an objective function where nonparametric parts have been concentrated out. That is, suppose that there is a function $q(z,\beta,\zeta)$ such that $m(z,\beta,\gamma)=\partial q(z,\beta,\zeta
(\beta))/\partial\beta$ where $\zeta(\beta)=\arg\max_{\zeta}E[q(z_{i},\beta,\zeta)]$, where $\gamma$ includes $\zeta(\beta)$ and possibly additional functions. Proposition 2 of Newey (1994a) and Lemma 2.5 of Chernozhukov et al. (2018) then imply that $m(z,\beta,\gamma)$ will be LR. This class of moment functions includes various partially linear regression models where $\zeta$ represents a conditional expectation. It also includes the efficient score for a semiparametric model, Newey (1994a, pp. 1358-1359).
Cross fitting, also known as sample splitting, has often been used to improve the properties of semiparametric and machine learning estimators; e.g. see Bickel (1982), Schick (1986), and Powell, Stock, and Stoker (1989). Cross fitting removes a source of bias and can be used to construct estimators with remainder terms that converge to zero as fast as is known to be possible, as in NHR and Newey and Robins (2017). Cross fitting is also useful for double machine learning estimators, as outlined in Chernozhukov et al. (2017, 2018). For these reasons we allow for cross-fitting, where sample moments have the form$$\hat{\psi}(\beta)=\frac{1}{n}\sum_{i=1}^{n}\psi(z_{i},\beta,\hat{\gamma}_{i},\hat{\lambda}_{i}),$$ with $\hat{\gamma}_{i}$ and $\hat{\lambda}_{i}$ being formed from observations other than the $i^{th}.$ This kind of cross fitting removes an “own observation” bias term and is useful for showing root-n consistency when $\hat{\gamma}_{i}$ and $\hat{\lambda}_{i}$ are machine learning estimators.
One version of cross-fitting with good properties in examples in Chernozhukov et al. (2018) can be obtained by partitioning the observation indices into $L$ groups $I_{\ell},(\ell=1,...,L),$ forming $\hat{\gamma}_{\ell}$ and $\hat{\lambda}_{\ell}$ from observations not in $I_{\ell}$, and constructing$$\hat{\psi}(\beta)=\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in I_{\ell}}\psi
(z_{i},\beta,\hat{\gamma}_{\ell},\hat{\lambda}_{\ell}). \label{cfit}$$ Further bias reductions may be obtained in some cases by using different sets of observations for computing $\hat{\gamma}_{\ell}$ and $\hat{\lambda}_{\ell
},$ leading to remainders that converge to zero as rapidly as known possible in interesting cases; see Newey and Robins (2017). The asymptotic theory of Section 7 focuses on this kind of cross fitting.
As an example we consider a bound on average equivalent variation. Let $\gamma_{0}(x)$ denote the conditional expectation of quantity $q$ conditional on $x=(p^{T},y)$ where $p=(p_{1},p_{2}^{T})^{T}$ is a vector of prices and $y$ is income$.$ The object of interest is a bound on average equivalent variation for a price change from $\bar{p}_{1}$ to $\check{p}_{1}$ given by$$\beta_{0}=E[\int\ell(p_{1},y_{i})\gamma_{0}(p_{1},p_{2i},y_{i})dp_{1}],\ell(p_{1},y)=w(y)1(\bar{p}_{1}\leq p_{1}\leq\check{p}_{1})\exp
\{-B(p_{1}-\bar{p}_{1})\}],$$ where $w(y)$ is a function of income and $B$ a constant. It follows by Hausman and Newey (2016) that if $B$ is a lower (upper) bound on the income effect for all individuals then $\beta_{0}$ is an upper (lower) bound on the equivalent variation for a price change from $\bar{p}_{1}$ to $\check{p}_{1},$ averaged over heterogeneity, other prices $p_{2i},$ and income $y_{i}$. The function $w(y)$ allows for averages over income in specific ranges, as in Hausman and Newey (2017).
A moment function that could be used to estimate $\beta_{0}$ is$$m(z,\beta,\gamma)=\int\ell(p_{1},y)\gamma(p_{1},p_{2},y)dp_{1}-\beta.$$ Note that $$E[m(z_{i},\beta_{0},\gamma)]+\beta_{0}=E[\int\ell(p_{1},y_{i})\gamma
(p_{1},p_{2i},y_{i})dp_{1}]=E[\lambda_{0}(x_{i})\gamma(x_{i})],\lambda
_{0}(x)=\frac{\ell(p_{1},y)}{f_{0}(p_{1}|p_{2},y)},$$ where $f_{0}(p_{1}|p_{2},y)$ is the conditional pdf of $p_{1i}$ given $p_{2i}$ and $y_{i}$. Then by Proposition 4 of Newey (1994) the influence function adjustment for any nonparametric estimator $\hat{\gamma}(x)$ of $E[q_{i}|x_{i}=x]$ is$$\phi(z,\beta,\gamma,\lambda)=\lambda(x)[q-\gamma(x)].$$ Here $\lambda_{0}(x)$ is an example of an additional unknown function that is included in $\phi(z,\beta,\gamma,\lambda)$ but not in the original moment functions $m(z,\beta,\gamma)$. Let $\hat{\gamma}_{i}(x)$ be an estimator of $E[q_{i}|x_{i}=x]$ that can depend on $i$ and $\hat{\lambda}_{i}(x)$ be an estimator of $\lambda_{0}(x)$, such as $\hat{f}_{i}(p_{1}|p_{2},y)^{-1}\ell(p_{1},y)$ for an estimator $\hat{f}_{i}(p_{1}|p_{2},y).$ The LR estimator obtained by solving $\hat{\psi}(\beta)=0$ for $m(z,\beta,\gamma)$ and $\phi(z,\beta,\gamma,\lambda)$ as above is$$\hat{\beta}=\frac{1}{n}\sum_{i=1}^{n}\left\{ \int\ell(p_{1},y_{i})\hat
{\gamma}_{i}(p_{1},p_{2i},y_{i})dp_{1}+\hat{\lambda}_{i}(x_{i})[q_{i}-\hat{\gamma}_{i}(x_{i})]\right\} . \label{exlr}$$
Machine Learning for Dynamic Discrete Choice
============================================
A challenging problem when estimating dynamic structural models is the dimensionality of state spaces. Machine learning addresses this problem via model selection to estimate high dimensional choice probabilities. These choice probabilities estimators can then be used in conditional choice probability (CCP) estimators of structural parameters, following Hotz and Miller (1993). In order for CCP estimators based on machine learning to be root-n consistent they must be based on orthogonal (i.e. LR) moment conditions, see Chernozhukov et al. (2017, 2018). Adding the adjustment term provides the way to construct LR moment conditions from known moment conditions for CCP estimators. In this Section we do so for the Rust’s (1987) model of dynamic discrete choice.
We consider an agent choosing among $J$ discrete alternatives by maximizing the expected present discounted value of utility. We assume that the per-period utility function for an agent making choice $j$ in period $t$ is given by$$U_{jt}=u_{j}(x_{t},\beta_{0})+\epsilon_{jt},(j=1,...,J;t=1,2,...).$$ The vector $x_{t}$ is the observed state variables of the problem (*e.g.* work experience, number of children, wealth) and the vector $\beta$ is unknown parameters. The disturbances $\epsilon_{t}=\{\epsilon
_{1t},...,\epsilon_{Jt}\}$ are not observed by the econometrician. As in much of the literature we assume that $\epsilon_{t}$ is i.i.d. over time with known CDF that has support $R^{J},$ is independent of $x_{t},$ and $x_{t}$ is first-order Markov.
To describe the agent’s choice probabilities let $\delta$ denote a time discount parameter, $\bar{v}(x)$ the expected value function, $y_{jt}\in\{0,1\}$ the indicator that choice $j$ is made and $\bar{v}_{j}(x_{t})=u_{j}(x_{t},\beta_{0})+\delta E[\bar{v}(x_{t+1})|x_{t},j]$ the expected value function conditional on choice $j.$ As in Rust (1987), we assume that in each period the agent makes the choice $j$ that maximizes the expected present discounted value of utility $\bar{v}_{j}(x_{t})+\epsilon
_{jt}.$ The probability of choosing $j$ in period $t$ is then$$P_{j}(\bar{v}_{t})=\Pr(\bar{v}_{j}(x_{t})+\epsilon_{jt}\geq\bar{v}_{k}(x_{t})+\epsilon_{kt};k=1,...,J),\bar{v}_{t}=(\bar{v}_{1}(x_{t}),...,\bar
{v}_{J}(x_{t}))^{\prime}. \label{choice prob}$$
These choice probabilities have a useful relationship to the structural parameters $\beta$ when there is a renewal choice, where the conditional distribution of $x_{t+1}$ given the renewal choice and $x_{t}$ does not depend on $x_{t}.$ Without loss of generality suppose that the renewal choice is $j=1.$ Let $\tilde{v}_{jt}$ denote $\tilde{v}_{j}(x_{t})=\bar{v}_{j}(x_{t})-\bar{v}_{1}(x_{t}),$ so that $\tilde{v}_{1t}\equiv0$. As usual, subtracting $\bar{v}_{1t}$ from each $\bar{v}_{jt}$ in $P_{j}(\bar{v}_{t})$ does not change the choice probabilities, so that they depend only on $\tilde{v}_{t}=(\tilde{v}_{2t},...,\tilde{v}_{Jt}).$
The renewal nature of $j=1$ leads to a specific formula for $\tilde{v}_{jt}$ in terms of the per period utilities $u_{jt}=u_{j}(x_{t},\beta_{0})$ and the choice probabilities $P_{t}=P(\tilde{v}_{t})=(P_{1}(\bar{v}_{t}),...P_{J}(\bar{v}_{t}))^{\prime}.$ As in Hotz and Miller (1993), there is a function $\mathcal{P}^{-1}(P)$ such that $\tilde{v}_{t}=\mathcal{P}^{-1}(P_{t}).$ Let $H(P)$ denote the function such that $$H(P_{t})=E[\max_{1\leq j\leq J}\{\mathcal{P}^{-1}(P_{t})_{j}+\epsilon
_{jt}\}|x_{t}]=E[\max_{1\leq j\leq J}\{\tilde{v}_{jt}+\epsilon_{jt}\}|x_{t}].$$ For example, for multinomial logit $H(P_{t})=.5772-\ln(P_{1t}).$ Note that by $j=1$ being a renewal we have $E[\bar{v}_{t+1}|x_{t},1]=C$ for a constant $C$, so that$$\bar{v}(x_{t})=\bar{v}_{1t}+H(P_{t})=u_{1t}+\delta C+H(P_{t}).$$ It then follows that$$\bar{v}_{jt}=u_{jt}+\delta E[\bar{v}(x_{t+1})|x_{t},j]=u_{jt}+\delta
E[u_{1,t+1}+H(P_{t+1})|x_{t},j]+\delta^{2}C,(j=1,...,J).$$ Subtracting then gives$$\tilde{v}_{jt}=u_{jt}-u_{1t}+\delta\{E[u_{1,t+1}+H(P_{t+1})|x_{t},j]-E[u_{1,t+1}+H(P_{t+1})|1]\}. \label{value}$$ This expression for the choice specific value function $\tilde{v}_{jt}$ depends only on $u_{j}(x_{t},\beta),$ $H(P_{t+1})$, and conditional expectations given the state and choice, and so can be used to form semiparametric moment functions.
To describe those moment functions let $\gamma_{1}(x)$ denote the vector of possible values of the choice probabilities $E[y_{t}|x_{t}=x],$ where $y_{t}=(y_{1t},...,y_{Jt})^{\prime}.$ Also let $\gamma_{j}(x_{t},\beta
,\gamma_{1}),(j=2,...,J)$ denote a possible $E[u_{1}(x_{t+1},\beta
)+H(\gamma_{1}(x_{t+1}))|x_{t},j]$ as a function of $\beta$, $x_{t}$ and $\gamma_{1},$ and $\gamma_{J+1}(\beta,\gamma_{1})$ a possible value of $E[u_{1}(x_{t},\beta)+H(\gamma_{1}(x_{t+1}))|1].$ Then a possible value of $\tilde{v}_{jt}$ is given by $$\tilde{v}_{j}(x_{t},\beta,\gamma)=u_{j}(x_{t},\beta)-u_{1}(x_{t},\beta
)+\delta\lbrack\gamma_{j}(x_{t},\beta,\gamma_{1})-\gamma_{J+1}(\beta
,\gamma_{1})],(j=2,...,J).$$ These value function differences are semiparametric, depending on the function $\gamma_{1}$ of choice probabilities and the conditional expectations $\gamma_{j}$, $(j=2,...,J).$ Let $\tilde{v}(x_{t},\beta,\gamma)=(\tilde{v}_{2}(x_{t},\beta,\gamma),...,\tilde{v}_{J}(x_{t},\beta,\gamma))^{\prime}$ and $A(x_{t})$ denote a matrix of functions of $x_{t}$ with $J$ columns. Semiparametric moment functions are given by$$m(z,\beta,\gamma)=A(x)[y-P(\tilde{v}(x,\beta,\gamma))].$$
LR moment functions can be constructed by adding the adjustment term for the presence of the first step $\gamma.$ This adjustment term is derived in Appendix A. It takes the form $$\phi(z,\beta,\gamma,\lambda)=\sum_{j=1}^{J+1}\phi_{j}(z,\beta,\gamma
,\lambda),$$ where $\phi_{j}(z,\beta,\gamma,\lambda)$ is the adjustment term for $\gamma_{j}$ holding all other components $\gamma$ fixed at their true values. To describe it define$$\begin{aligned}
P_{\tilde{v}j}(\tilde{v}) & =\partial P(\tilde{v})/\partial\tilde{v}_{j},\text{ }\pi_{1}=\Pr(y_{t1}=1),\text{ }\lambda_{10}(x)=E[y_{1t}|x_{t+1}=x],\label{ddcdef}\\
\lambda_{j0}(x) & =E[A(x_{t})P_{\tilde{v}j}(\tilde{v}_{t})\frac{y_{tj}}{P_{j}(\tilde{v}_{t})}|x_{t+1}=x],(j=2,...,J).\nonumber\end{aligned}$$ Then for $w_{t}=x_{t+1}$ and $z=(y,x,w)$ let$$\begin{aligned}
\phi_{1}(z,\beta,\gamma,\lambda) & =-\delta\left( \sum_{j=2}^{J}\{\lambda_{j}(x)-E[A(x_{t})P_{\tilde{v}j}(\tilde{v}_{t})]\pi_{1}^{-1}\lambda_{1}(x)\}\right) [\partial H(\gamma_{1}(x))/\partial P]^{\prime
}\{y-\gamma_{1}(x)\}\\
\phi_{j}(z,\beta,\gamma,\lambda) & =-\delta A(x)P_{\tilde{v}j}(\tilde
{v}(x,\beta,\gamma))\frac{y_{j}}{P_{j}(\tilde{v}(x,\beta,\gamma))}\{u_{1}(w,\beta)+H(\gamma_{1}(w))-\gamma_{j}(x,\beta,\gamma_{1})\},(j=2,...,J),\\
\phi_{J+1}(z,\beta,\gamma,\lambda) & =\delta\left( \sum_{j=2}^{J}E[A(x_{t})P_{\tilde{v}j}(\tilde{v}(x_{t},\beta,\gamma))]\right) \pi_{1}^{-1}y_{1}\{u_{1}(w,\beta)+H(\gamma_{1}(w))-\gamma_{J+1}(\beta,\gamma_{1})\}.\end{aligned}$$
<span style="font-variant:small-caps;">Theorem 2:</span> *If the marginal distribution of* $x_{t}$ *does not vary with* $t$ *then LR moment functions for the dynamic discrete choice model are*$$\psi(z,\beta,\gamma)=A(x_{t})[y_{t}-P(\tilde{v}(x_{t},\beta,\gamma
))]+\sum_{j=1}^{J+1}\phi_{j}(z,\beta,\lambda).$$
The form of $\psi(z,\beta,\gamma)$ is amenable to machine learning. A machine learning estimator of the conditional choice probability vector $\gamma
_{10}(x)$ is straightforward to compute and can then be used throughout the construction of the orthogonal moment conditions everywhere $\gamma_{1}$ appears. If $u_{1}(x,\beta)$ is linear in $x,$ say $u_{1}(x,\beta
)=x_{1}^{\prime}\beta_{1}$ for subvectors $x_{1}$ and $\beta_{1}$ of $x$ and $\beta$ respectively, then machine learning estimators can be used to obtain $\hat{E}[x_{1,t+1}|x_{t},j]$ and $\hat{E}[\hat{H}_{t+1}|x_{j},j],$ $(j=2,...,J),$ and a sample average used to form $\hat{\gamma}_{J+1}(\beta,\hat{\gamma}_{1})$. The value function differences can then be estimated as$$\tilde{v}_{j}(x_{t},\beta,\hat{\gamma})=u_{j}(x_{t},\beta)-u_{1}(x_{t},\beta)+\hat{E}[x_{1,t+1}|x_{t},j]^{\prime}\beta_{1}-\hat{E}[x_{1,t+1}|1]^{\prime}\beta_{1}+\hat{E}[\hat{H}_{t+1}|x_{t},j]-\hat{E}[\hat{H}_{t+1}|1].$$ Furthermore, denominator problems can be avoided by using structural probabilities (rather than the machine learning estimators) in all denominator terms.
The challenging part of the machine learning for this estimator is the dependence on $\beta$ of the reverse conditional expectations in $\lambda
_{1}(x)$. It may be computationally prohibitive and possibly unstable to redo machine learning for each $\beta.$ One way to to deal with this complication is to update $\beta$ periodically, with more frequent updates near convergence. It is important that at convergence the $\beta$ in the reverse conditional expectations is the same as the $\beta$ that appears elsewhere.
With data $z_{i}$ that is i.i.d. over individuals these moment functions can be used for any $t$ to estimate the structural parameters $\beta.$ Also, for data for a single individual we could use a time average $\sum_{t=1}^{T-1}\psi(z_{t},\beta,\gamma)/(T-1)$ to estimate $\beta.$ It will be just as important to use LR moments for estimation with a single individual as it is with a cross section of individuals, although our asymptotic theory will not apply to that case.
Bajari, Chernozhukov, Hong, and Nekipelov (2009) derived the influence adjustment for dynamic discrete games of imperfect information. Locally robust moment conditions for such games could be formed using their results. We leave that formulation to future work.
As an example of the finite sample performance of the LR GMM we report a Monte Carlo study of the LR estimator of this Section. The design of the experiment is loosely like the bus replacement application of Rust (1987). Here $x_{t}$ is a state variable meant to represent the lifetime of a bus engine. The transition density is $$x_{t+1}=\left\{
\begin{array}
[c]{c}x_{t}+N(.25,1)^{2},y_{t}=1,\\
x_{t}=1+N(.25,1)^{2},y_{t}=0.
\end{array}
\right. .$$ where $y_{t}=0$ corresponds to replacement of the bus engine and $y_{t}=1$ to nonreplacement. We assume that the agent chooses $y_{t}$ contingent on state to maximize$$\sum_{t=1}^{\infty}\delta^{t-1}[y_{t}(\alpha\sqrt{x_{t}}+\varepsilon
_{t})+(1-y_{t})RC],\alpha=-.3,RC=-4.$$ The unconditional probability of replacement in this model is about $1/8,$ which is substantially higher than that estimated in Rust (1987). The sample used for estimation was $1000$ observations for a single decision maker. We carried out $10,000$ replications.
We estimate the conditional choice probabilities by kernel and series nonparametric regression and by logit lasso, random forest, and boosted tree machine learning methods. Logit conditional choice probabilities and derivatives were used in the construction of $\hat{\lambda}_{j}$ wherever they appear in order to avoid denominator issues. The unknown conditional expectations in the $\hat{\lambda}_{j}$ were estimated by series regressions throughout. Kernel regression was also tried but did not work particularly well and so results are not reported.
Table 1 reports the results of the experiment. Bias, standard deviation, and coverage probability for asymptotic 95 percent confidence intervals are given in Table 1.
Table 1
\[c\][lllllll]{}\
& & &\
& $\ \ \ \ \alpha$ & RC & $\ \ \ \alpha$ & RC & $\ \ \ \alpha$ & RC\
Two step kernel & -.24 & .08 & .08 & .32 & .01 & .86\
LR kernel & -.05 & .02 & .06 & .32 & .95 & .92\
Two step quad & -.00 & .14 & .049 & .33$^{\ast}$ & .91 & .89\
LR quad & -.00 & .01 & .085 & .39 & .95 & .92\
Logit Lasso & -.12 & .25 & .06 & .28 & .74 & .84\
LR Logit Lasso & -.09 & .01 & .08 & .36 & .93 & .95\
Random Forest & -.15 & -.44 & .09 & .50 & .91 & .98\
LR Ran. For. & .00 & .00 & .06 & .44 & 1.0 & .98\
Boosted Trees & -.10 & -.28 & .08 & .50 & .99 & .99\
LR Boost Tr. & .03 & .09 & .07 & .47 & .99 & .97
Here we find bias reduction from the LR estimator in all cases. We also find variance reduction from LR estimation when the first step is kernel estimation, random forests, and boosted trees. The LR estimator also leads to actual coverage of confidence intervals being closer to the nominal coverage. The results for random forests and boosted trees seem noisier than the others, with higher standard deviations and confidence interval coverage probabilities farther from nominal. Overall, we find substantial improvements from using LR moments rather than only the identifying, original moments.
Estimating the Influence Adjustment
===================================
Construction of LR moment functions requires an estimator $\hat{\phi}(z,\beta)$ of the adjustment term. The form of $\phi(z,\beta,\gamma,\lambda)$ is known for some cases from the semiparametric estimation literature. Powell, Stock, and Stoker (1989) derived the adjustment term for density weighted average derivatives. Newey (1994a) gave the adjustment term for mean square projections (including conditional expectations), densities, and their derivatives. Hahn (1998) and Hirano, Imbens, and Ridder (2003) used those results to obtain the adjustment term for treatment effect estimators, where the LR estimator will be the doubly robust estimator of Robins, Rotnitzky, and Zhao (1994, 1995). Bajari, Hong, Krainer, and Nekipelov (2010) and Bajari, Chernozhukov, Hong, and Nekipelov (2009) derived adjustment terms in some game models. Hahn and Ridder (2013, 2016) derived adjustments in models with generated regressors including control functions. These prior results can be used to obtain LR estimators by adding the adjustment term with nonparametric estimators plugged in.
For new cases it may be necessary to derive the form of the adjustment term. Also, it is possible to numerically estimate the adjustment term based on series estimators and other nonparametric estimators. In this Section we describe how to construct estimators of the adjustment term in these ways.
Deriving the Formula for the Adjustment Term
--------------------------------------------
One approach to estimating the adjustment term is to derive a formula for $\phi(z,\beta,\gamma,\lambda)$ and then plug in $\hat{\gamma}$ and $\hat{\lambda}$ in that formula$.$ A formula for $\phi(z,\beta,\gamma
,\lambda)$ can be obtained as in Newey (1994a). Let $\gamma(F)$ be the limit of the nonparametric estimator $\hat{\gamma}$ when $z_{i}$ has distribution $F.$ Also, let $F_{\tau}$ denote a regular parametric model of distributions with $F_{\tau}=F_{0}$ at $\tau=0$ and score (derivative of the log likelihood at $\tau=0)$ equal to $S(z)$. Then under certain regularity conditions $\phi(z,\beta,\gamma_{0},\lambda_{0})$ will be the unique solution to$$\left. \frac{\partial\int m(z,\beta,\gamma(F_{\tau}))F_{0}(dz)}{\partial\tau
}\right\vert _{\tau=0}=E[\phi(z_{i},\beta,\gamma_{0},\lambda_{0})S(z_{i})],E[\phi(z_{i},\beta,\gamma_{0},\lambda_{0})]=0, \label{funeq}$$ as $\{F_{\tau}\}$ and the corresponding score $S(z)$ are allowed to vary over a family of parametric models where the set of scores for the family has mean square closure that includes all mean zero functions with finite variance. Equation (\[funeq\]) is a functional equation that can be solved to find the adjustment term, as was done in many of the papers cited in the previous paragraph.
The influence adjustment can be calculated by taking a limit of the Gateaux derivative as shown in Ichimura and Newey (2017). Let $\gamma(F)$ be the limit of $\hat{\gamma}$ when $F$ is the true distribution of $z_{i}$, as before. Let $G_{z}^{h}$ be a family of distributions that approaches a point mass at $z$ as $h\longrightarrow0.$ If $\phi(z_{i},\beta,\gamma_{0},\lambda_{0})$ is continuous in $z_{i}$ with probability one then$$\phi(z,\beta,\gamma_{0},\lambda_{0})=\lim_{h\longrightarrow0}\left( \left.
\frac{\partial E[m(z_{i},\beta,\gamma(F_{\tau}^{h}))]}{\partial\tau
}\right\vert _{\tau=0}\right) ,F_{\tau}^{h}=(1-\tau)F_{0}+\tau G_{z}^{h}.
\label{derlim}$$ This calculation is more constructive than equation (\[funeq\]) in the sense that the adjustment term here is a limit of a derivative rather than the solution to a functional equation. In Sections 5 and 6 we use those results to construct LR estimators when the first step is a nonparametric instrumental variables (NPIV) estimator.
With a formula for $\phi(z,\beta,\gamma,\lambda)$ in hand from either solving the functional equation in equation (\[funeq\]) or from calculating the limit of the derivative in equation (\[derlim\]), one can estimate the adjustment term by plugging estimators $\hat{\gamma}$ and $\hat{\lambda}$ into $\phi(z,\beta,\gamma,\lambda).$ This approach to estimating LR moments can used to construct LR moments for the average surplus described near the end of Section 2. There the adjustment term depends on the conditional density of $p_{1i}$ given $p_{2i}$ and $y_{i}$. Let $\hat{f}_{\ell}(p_{1}|p_{2},y)$ be some estimator of the conditional pdf of $p_{1i}$ given $p_{2i}$ and $y_{i}.$ Plugging that estimator into the formula for $\lambda_{0}(x)$ gives $\hat{\lambda}_{\ell}(x)=\frac{\ell(p_{1},y)}{\hat{f}_{\ell}(p_{1}|p_{2},y)}.$This $\hat{\lambda}_{\ell}(x)$ can then be used in equation (\[exlr\])$.$
Estimating the Influence Adjustment for First Step Series Estimators
--------------------------------------------------------------------
Estimating the adjustment term is relatively straightforward when the first step is a series estimator. The adjustment term can be estimated by treating the first step estimator as if it were parametric and applying a standard formula for the adjustment term for parametric two-step estimators. Suppose that $\hat{\gamma}_{\ell}$ depends on the data through a $K\times1$ vector $\hat{\zeta}_{\ell}$ of parameter estimators that has true value $\zeta_{0}$. Let $m(z,\beta,\zeta)$ denote $m(z,\beta,\gamma)$ as a function of $\zeta.$ Suppose that there is a $K\times1$ vector of functions $h(z,\zeta)$ such that $\hat{\zeta}_{\ell}$ satisfies$$\frac{1}{\sqrt{\bar{n}_{\ell}}}\sum_{i\in\bar{I}_{\ell}}h(z_{i},\hat{\zeta
}_{\ell})=o_{p}(1),$$ where $\bar{I}_{\ell}$ is a subset of observations, none which are included in $I_{\ell},$ and $\bar{n}_{\ell}$ is the number of observations in $\bar
{I}_{\ell}.$ Then a standard calculation for parametric two-step estimators (e.g. Newey, 1984, and Murphy and Topel, 1985) gives the parametric adjustment term$$\phi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell})=\hat{\Psi}_{\ell}(\beta)h(z_{i},\hat{\zeta}_{\ell}),\hat{\Psi}_{\ell}(\beta)=-\sum_{j\in\bar
{I}_{\ell}}\frac{\partial m(z_{j},\beta,\hat{\zeta}_{\ell})}{\partial\zeta
}\left( \sum_{j\in\bar{I}_{\ell}}\frac{\partial h(z_{j},\hat{\zeta}_{\ell})}{\partial\zeta}\right) ^{-1},i\in I_{\ell}.$$ In many cases $\phi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell})$ approximates the true adjustment term $\phi(z,\beta,\gamma_{0},\lambda_{0}),$ as shown by Newey (1994a, 1997) and Ackerberg, Chen, and Hahn (2012) for estimating the asymptotic variance of functions of series estimators. Here this approximation is used for estimation of $\beta$ instead of just for variance estimation. The estimated LR moment function will be$$\psi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell})=m(z_{i},\beta
,\hat{\zeta}_{\ell})+\phi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell}).
\label{lr series}$$ We note that if $\hat{\zeta}_{\ell}$ were computed from the whole sample then $\hat{\phi}(\beta)=0$. This degeneracy does not occur when cross-fitting is used, which removes “own observation” bias and is important for first step machine learning estimators, as noted in Section 2.
We can apply this approach to construct LR moment functions for an estimator of the average surplus bound example that is based on series regression. Here the first step estimator of $\gamma_{0}(x)=E[q_{i}|x_{i}=x]$ will be that from an ordinary least regression of $q_{i}$ on a vector $a(x_{i})$ of approximating functions. The corresponding $m(z,\beta,\zeta)$ and $h(z,\zeta)$ are$$m(z,\beta,\zeta)=A(x)^{\prime}\zeta-\beta,h(z,\zeta)=a(x)[q-a(x)^{\prime}\zeta],A(x)=\int\ell(p_{1},y)a(p_{1},p_{2},y)dp_{1}.$$ Let $\hat{\zeta}_{\ell}$ denote the least squares coefficients from regressing $q_{i}$ on $a(x_{i})$ for observations that are not included in $I_{\ell}$. Then the estimator of the locally robust moments given in equation (\[lr series\]) is $$\begin{aligned}
\psi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell}) & =A(x_{i})^{\prime
}\hat{\zeta}_{\ell}-\beta+\hat{\Psi}_{\ell}a(x_{i})[q_{i}-a(x_{i})^{\prime
}\hat{\zeta}_{\ell}],\\
\hat{\Psi}_{\ell} & =\sum_{j\in\bar{I}_{\ell}}A(x_{j})^{\prime}\left(
\sum_{j\in\bar{I}_{\ell}}a(x_{j})a(x_{j})^{\prime}\right) ^{-1}.\end{aligned}$$ It can be shown similarly to Newey (1994a, p. 1369) that $\hat{\Psi}_{\ell}$ estimates the population least squares coefficients from a regression of $\lambda_{0}(x_{i})$ on $a(x_{i}),$ so that $\hat{\lambda}_{\ell}(x_{i})=\hat{\Psi}_{\ell}a(x_{i})$ estimates $\lambda_{0}(x_{i}).$ In comparison the LR estimator described in the previous subsection was based on an explicit nonparametric estimator of $f_{0}(p_{1}|p_{2},y),$ while this $\hat{\lambda
}_{\ell}(x)$ implicitly estimates the inverse of that pdf via a mean-square approximation of $\lambda_{0}(x_{i})$ by $\hat{\Psi}_{\ell}a(x_{i}).$
Chernozhukov, Newey, and Robins (2018) introduce machine learning methods for choosing the functions to include in the vector $A(x)$. This method can be combined with machine learning methods for estimating $E[q_{i}|x_{i}]$ to construct a double machine learning estimator of average surplus, as shown in Chernozhukov, Hausman, and Newey (2018).
In parametric models moment functions like those in equation (\[lr series\]) are used to “partial out” nuisance parameters $\zeta.$ For maximum likelihood these moment functions are the basis of Neyman’s (1959) C-alpha test. Wooldridge (1991) generalized such moment conditions to nonlinear least squares and Lee (2005), Bera et al. (2010), and Chernozhukov et al. (2015) to GMM. What is novel here is their use in the construction of semiparametric estimators and the interpretation of the estimated LR moment functions $\psi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell})$ as the sum of an original moment function $m(z_{i},\beta,\hat{\zeta}_{\ell})$ and an influence adjustment $\phi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell})$.
Estimating the Influence Adjustment with First Step Smoothing
-------------------------------------------------------------
The adjustment term can be estimated in a general way that allows for kernel density, locally linear regression, and other kernel smoothing estimators for the first step. The idea is to differentiate with respect to the effect of the $i^{th}$ observation on sample moments. Newey (1994b) used a special case of this approach to estimate the asymptotic variance of a functional of a kernel based semiparametric or nonparametric estimator. Here we extend this method to a wider class of first step estimators, such as locally linear regression, and apply it to estimate the adjustment term for construction of LR moments.
We will describe this estimator for the case where $\gamma$ is a vector of functions of a vector of variables $x.$ Let $h(z,x,\gamma)$ be a vector of functions of a data observation $z$, $x$, and a possible realized value of $\gamma$ (i.e. a vector of real numbers $\gamma$). Also let $\hat{h}_{\ell
}(x,\gamma)=\sum_{j\in\bar{I}_{\ell}}h(z_{j},x,\gamma)/\bar{n}_{\ell}$ be a sample average over a set of observations $\bar{I}_{\ell}$ not included in $I_{\ell},$ where $\bar{n}_{j}$ is the number of observations in $\bar{I}_{j}.$ We assume that the first step estimator $\hat{\gamma}_{\ell}(x)$ solves$$0=\hat{h}_{\ell}(x,\gamma).$$ We suppress the dependence of $h$ and $\hat{\gamma}$ on a bandwidth. For example for a pdf $\kappa(u)$ a kernel density estimator would correspond to $h(z_{j},x,\gamma)=\kappa(x-x_{j})-\gamma$ and a locally linear regression would be $\hat{\gamma}_{1}(x)$ for$$h(z_{j},x,\gamma)=\kappa(x-x_{j})\left(
\begin{array}
[c]{c}1\\
x-x_{j}\end{array}
\right) [y_{j}-\gamma_{1}-(x-x_{j})^{\prime}\gamma_{2}].$$
To measure the effect of the $i^{th}$ observation on $\hat{\gamma}$ let $\hat{\gamma}_{\ell i}^{\xi}(x)$ be the solution to $$0=\hat{h}_{\ell}(x,\gamma)+\xi\cdot h(z_{i},x,\gamma).$$ This $\hat{\gamma}_{\ell i}^{\xi}(x)$ is the value of the function obtained from adding the contribution $\xi\cdot h(z_{i},x,\gamma)$ of the $i^{th}$ observation. An estimator of the adjustment term can be obtained by differentiating the average of the original moment function with respect to $\xi$ at $\xi=0.$ This procedure leads to an estimated locally robust moment function given by$$\psi(z_{i},\beta,\hat{\gamma}_{\ell})=m(z_{i},\beta,\hat{\gamma}_{\ell
})+\left. \frac{\partial}{\partial\xi}\frac{1}{\bar{n}_{\ell}}\sum_{j\in
\bar{I}_{\ell}}m(z_{j},\beta,\hat{\gamma}_{\ell i}^{\xi}(\cdot))\right\vert
_{\xi=0}.$$ This estimator is a generalization of the influence function estimator for kernels in Newey (1994b).
Double and Partial Robustness
=============================
The zero derivative condition in equation (\[lrdef\]) is an appealing robustness property in and of itself. A zero derivative means that the expected moment functions remain closer to zero than $\tau$ as $\tau$ varies away from zero. This property can be interpreted as local insensitivity of the moments to the value of $\gamma$ being plugged in, with the moments remaining close to zero as $\gamma$ varies away from its true value. Because it is difficult to get nonparametric functions exactly right, especially in high dimensional settings, this property is an appealing one.
Such robustness considerations, well explained in Robins and Rotnitzky (2001), have motivated the development of doubly robust (DR) moment conditions. DR moment conditions have expectation zero if one first stage component is incorrect. DR moment conditions allow two chances for the moment conditions to hold, an appealing robustness feature. Also, DR moment conditions have simpler conditions for asymptotic normality than general LR moment functions as discussed in Section 7. Because many interesting LR moment conditions are also DR we consider double robustness.
LR moments that are constructed by adding the adjustment term for first step estimation provide candidates for DR moment functions. The derivative of the expected moments with respect to each first step will be zero, a necessary condition for DR. The condition for moments constructed in this way to be DR is the following:
<span style="font-variant:small-caps;">Assumption 1:</span> *There are sets* $\Gamma$ *and* $\Lambda
$ *such that for all* $\gamma\in\Gamma$ *and* $\lambda\in
\Lambda$$$E[m(z_{i},\beta_{0},\gamma)]=-E[\phi(z_{i},\beta_{0},\gamma,\lambda
_{0})],E[\phi(z_{i},\beta_{0},\gamma_{0},\lambda)]=0.$$
This condition is just the definition of DR for the moment function $\psi(z,\beta,\gamma)=m(z,\beta,\gamma)+\phi(z,\beta,\gamma,\lambda)$, pertaining to specific sets $\Gamma$ ** and $\Lambda.$
The construction of adding the adjustment term to an identifying or original moment function leads to several novel classes of DR moment conditions. One such class has a first step that satisfies a conditional moment restriction$$E[y_{i}-\gamma_{0}(w_{i})|x_{i}]=0, \label{cmrlin}$$ where $w_{i}$ is potentially endogenous and $x_{i}$ is a vector of instrumental variables. This condition is the nonparametric instrumental variable (NPIV) restriction as in Newey and Powell (1989, 2003) and Newey (1991). A first step conditional expectation where $\gamma_{0}(x_{i})=E[y_{i}|x_{i}]$ is included as special case with $w_{i}=x_{i}.$ Ichimura and Newey (2017) showed that the adjustment term for this step takes the form $\phi(z,\gamma,\lambda)=\lambda(x)[y-\gamma(w)]$ so $m(z,\beta,\gamma
)+\lambda(x)[y-\gamma(x)]$ is a candidate for a DR moment function. A sufficient condition for DR is:
<span style="font-variant:small-caps;">Assumption 2:</span> *i) Equation (\[cmrlin\]) is satisfied; ii)* $\Lambda=\{\lambda(x):E[\lambda(x_{i})^{2}]<\infty\}$ *and* $\Gamma=\{\gamma(w):E[\gamma(w_{i})^{2}]<\infty\};$ *iii) there is* $v(w)$ *with* $E[v(w_{i})^{2}]<\infty$ *such that* $E[m(z_{i},\beta_{0},\gamma)]=E[v(w_{i})\{\gamma(w_{i})-\gamma_{0}(w_{i})\}]$ *for all* $\gamma\in\Gamma$*; iv) there is* $\lambda
_{0}(x)$ *such that* $v(w_{i})=E[\lambda_{0}(x_{i})|w_{i}]$*; and v)* $E[y_{i}^{2}]<\infty.$
By the Riesz representation theorem condition iii) is necessary and sufficient for $E[m(z_{i},\beta_{0},\gamma)]$ to be a mean square continuous functional of $\gamma$ with representer $v(w).$ Condition iv) is an additional condition giving continuity in the reduced form difference $E[\gamma(w_{i})-\gamma
_{0}(w_{i})|x_{i}]$, as further discussed in Ichimura and Newey (2017). Under this condition$$\begin{aligned}
E[m(z_{i},\beta_{0},\gamma)] & =E[E[\lambda_{0}(x_{i})|w_{i}]\{\gamma
(w_{i})-\gamma_{0}(w_{i})\}]=E[\lambda_{0}(x_{i})\{\gamma(w_{i})-\gamma
_{0}(w_{i})\}]\\
& =-E[\phi(z_{i},\gamma,\lambda_{0})],\text{ \ }E[\phi(z_{i},\gamma
_{0},\lambda)]=E[\lambda(x_{i})\{y_{i}-\gamma_{0}(w_{i})\}]=0.\end{aligned}$$ Thus Assumption 2 implies Assumption 1 so that we have
<span style="font-variant:small-caps;">Theorem 3:</span> *If Assumption 2 is satisfied then* $m(z,\beta
,\gamma)+\lambda(x)\{y-\gamma(w)\}$ *is doubly robust.*
There are many interesting, novel examples of DR moment conditions that are special cases of Theorem 3. The average surplus bound is an example where $y_{i}=q_{i},$ $w_{i}=x_{i},$ $x_{i}$ is the observed vector of prices and income, $\Lambda=\Gamma$ is the set of all measurable functions of $x_{i}$ with finite second moment, and $\gamma_{0}(x)=E[y_{i}|x_{i}=x].$ Let $x_{1}$ denote $p_{1}$ and $x_{2}$ the vector of other prices and income, so that $x=(x_{1},x_{2}^{\prime})^{\prime}$. Also let $f_{0}(x_{1}|x_{2})$ denote the conditional pdf of $p_{1}$ given $x_{2}$ and $\ell(x)=\ell(p_{1},y)$ for income $y$. Let $m(z,\beta,\gamma)=\int\ell(p_{1},x_{2})\gamma(p_{1},x_{2})dp_{1}-\beta$ as before. Multiplying and dividing through by $f_{0}(p_{1}|x_{2})$ gives, for all $\gamma,\lambda\in\Gamma$ and $\lambda
_{0}(x)=f_{0}(x_{1}|x_{2})^{-1}\ell(x),$ $$E[m(z_{i},\beta_{0},\gamma)]=E[\int\ell(p_{1},x_{2i})\gamma(p_{1},x_{2i})dp_{1}]-\beta_{0}=E[E[\lambda_{0}(x_{i})\gamma(x_{i})|x_{2i}]]-\beta_{0}=E[\lambda_{0}(x_{i})\{\gamma(x_{i})-\gamma_{0}(x_{i})\}].$$ Theorem 3 then implies that the LR moment function for average surplus $m(z,\beta,\gamma)+\lambda(x)[q-\gamma(x)]$ is DR. A corresponding DR estimator $\hat{\beta}$ is given in equation (\[exlr\]).
The surplus bound is an example of a parameter where $\beta_{0}=E[g(z_{i},\gamma_{0})]$ for some linear functional $g(z,\gamma)$ of $\gamma$ and for $\gamma_{0}$ satisfying the conditional moment restriction of equation (\[cmrlin\])$.$ For the surplus bound $g(z,\gamma)=\int\ell(p_{1},x_{2})\gamma(p_{1},x_{2})dp_{1}.$ If Assumption 2 is satisfied then choosing $m(z,\beta,\gamma)=g(z,\gamma)-\beta$ a DR moment condition is $g(z,\gamma
)-\beta+\lambda(x)[y-\gamma(w)].$ A corresponding DR estimator is$$\hat{\beta}=\frac{1}{n}\sum_{i=1}^{n}\{g(z_{i},\hat{\gamma}_{i})+\hat{\lambda
}_{i}(x_{i})[y_{i}-\hat{\gamma}_{i}(w_{i})]\}, \label{drlin}$$ where $\hat{\gamma}_{i}(w)$ and $\hat{\lambda}_{i}(x)$ are estimators of $\gamma_{0}(w)$ and $\lambda_{0}(x)$ respectively. An estimator $\hat{\gamma
}_{i}$ can be constructed by nonparametric regression when $w_{i}=x_{i}$ or NPIV in general. A series estimator $\hat{\lambda}_{i}(x)$ can be constructed similarly to the surplus bound example in Section 3.2. For $w_{i}=x_{i}$ Newey and Robins (2017) give such series estimators of $\hat{\lambda}_{i}(x)$ and Chernozhukov, Newey, and Robins (2018) show how to choose the approximating functions for $\hat{\lambda}_{i}(x_{i})$ by machine learning. Simple and general conditions for root-n consistency and asymptotic normality of $\hat{\beta}$ that allow for machine learning are given in Section 7.
Novel examples of the DR estimator in equation (\[drlin\]) $w_{i}=x_{i}$ are given by Newey and Robins (2017) and Chernozhukov, Newey, and Robins (2018). Also Appendix C provides a generalization to $\gamma(w)$ and $\gamma(x)$ that satisfy orthogonality conditions more general than conditional moment restrictions and novel examples of those. A novel example with $w_{i}\neq
x_{i}$ is a weighted average derivative of $\gamma_{0}(w)$ satisfying equation (\[cmrlin\]). Here $g(z,\gamma)=\bar{v}(w)\partial\gamma(w)/\partial w$ for some weight function $\bar{v}(w)$. Let $f_{0}(w)$ be the pdf of $w_{i}$ and $v(w)=-f_{0}(w)^{-1}\partial\lbrack\bar{v}(w)f_{0}(w)]/\partial w,$ assuming that derivatives exist. Assume that $\bar{v}(w)\gamma(w)f_{0}(w)$ is zero on the boundary of the support of $w_{i}.$ Integration by parts then gives Assumption 2 iii). Assume also that there exists $\lambda_{0}\in\Lambda$ with $v(w_{i})=E[\lambda_{0}(x_{i})|w_{i}].$ Then for estimators $\hat{\gamma}_{i}$ and $\hat{\lambda}_{i}$ a DR estimator of the weighted average derivative is$$\hat{\beta}=\frac{1}{n}\sum_{i=1}^{n}\{\bar{v}(w_{i})\frac{\partial\hat
{\gamma}_{i}(w_{i})}{\partial w}+\hat{\lambda}_{i}(x_{i})[y_{i}-\hat{\gamma
}_{i}(w_{i})]\}.$$ This is a DR version of the weighted average derivative estimator of Ai and Chen (2007). A special case of this example is the DR moment condition for the weighted average derivative in the exogenous case where $w_{i}=x_{i}$ given in Firpo and Rothe (2017).
Theorem 3 includes existing DR moment functions as special cases where $w_{i}=x_{i}$, including the mean with randomly missing data given by Robins and Rotnitzky (1995), the class of DR estimators in Robins et al. (2008), and the DR estimators of Firpo and Rothe (2017). We illustrate for the mean with missing data. Let $w=x,$ $x=(a,u)$ for an observed data indicator $a\in\{0,1\}$ and covariates $u,$ $m(z,\beta,\gamma)=\gamma(1,u)-\beta,$ and $\lambda_{0}(x)=a/\Pr(a_{i}=1|u_{i}=u).$ Here it is well known that $$E[m(z_{i},\beta_{0},\gamma)]=E[\gamma(1,u_{i})]-\beta_{0}=E[\lambda_{0}(x_{i})\{\gamma(x_{i})-\gamma_{0}(x_{i})\}]=-E[\lambda_{0}(x_{i})\{y_{i}-\gamma(x_{i})\}].$$ Then DR of the moment function $\gamma(1,w)-\beta+\lambda(x)[y-\gamma(x)]$ of Robins and Rotnitzky (1995) follows by Proposition 5.
Another novel class of DR moment conditions are those where the first step $\gamma$ is a pdf of a function $x$ of the data observation $z.$ By Proposition 5 of Newey (1994a), the adjustment term for such a first step is $\phi(z,\beta,\gamma,\lambda)=\lambda(x)-\int\lambda(u)\gamma(u)du$ for some possible $\lambda$. A sufficient condition for the DR as in Assumption 1 is:
<span style="font-variant:small-caps;">Assumption 3:</span> $x_{i}$ *has pdf* $\gamma_{0}(x)$ *and for* $\Gamma=\{\gamma:\gamma(x)\geq0$, $\int\gamma(x)dx=1\}$ *there is* $\lambda_{0}(x)$ *such that for all* $\gamma\in\Gamma,$$$E[m(z_{i},\beta_{0},\gamma)]=\int\lambda_{0}(x)\{\gamma(x)-\gamma_{0}(x)\}dx.$$
Note that for $\phi(z,\gamma,\lambda)=\lambda(x)-\int\lambda(\tilde{x})\gamma(\tilde{x})d\tilde{x}$ it follows from Assumption 3 that $E[m(z_{i},\beta_{0},\gamma)]=-E[\phi(z_{i},\gamma,\lambda_{0})]$ for all $\gamma
\in\Gamma$. Also, $E[\phi(z_{i},\gamma_{0},\lambda)]=E[\lambda(x_{i})]-\int\lambda(\tilde{x})\gamma_{0}(\tilde{x})dx=0.$ Then Assumption 1 is satisfied so we have:
<span style="font-variant:small-caps;">Theorem 4:</span> *If Assumption 3 is satisfied then* $m(z,\beta
,\gamma)+\lambda(x)-\int\lambda(\tilde{x})\gamma(\tilde{x})d\tilde{x}$ *is DR.*
The integrated squared density $\beta_{0}=\int\gamma_{0}(x)^{2}dx$ is an example for $m(z,\beta,\gamma)=\gamma(x)-\beta,$ $\lambda_{0}=\gamma_{0},$ and $$\psi(z,\beta,\gamma,\lambda)=\gamma(x)-\beta+\lambda(x)-\int\lambda(\tilde
{x})\gamma(\tilde{x})dx.$$ This DR moment function seems to be novel. Another example is the density weighted average derivative (DWAD) of Powell, Stock, and Stoker (1989), where $m(z,\beta,\gamma)=-2y\cdot\partial\gamma(x)/\partial x-\beta$. Let $\delta(x_{i})=E[y_{i}|x_{i}]\gamma_{0}(x_{i})$. Assuming that $\delta
(u)\gamma(u)$ is zero on the boundary and differentiable, integration by parts gives$$E[m(z_{i},\beta_{0},\gamma)]=-2E[y_{i}\partial\gamma(x_{i})/\partial
x]-\beta_{0}=\int[\partial\delta(\tilde{x})/\partial x]\{\gamma(\tilde
{x})-\gamma_{0}(\tilde{x})\}du,$$ so that Assumption 3 is satisfied with $\lambda_{0}(x)=\partial\delta
(x)/\partial x.$ Then by Theorem 4$$\hat{\beta}=\frac{1}{n}\sum_{i=1}^{n}\{-2\frac{\partial\hat{\gamma}_{i}(x_{i})}{\partial x}+\frac{\partial\hat{\delta}_{i}(x_{i})}{\partial x}-\int\frac{\partial\hat{\delta}_{i}(\tilde{x})}{\partial x}\hat{\gamma}_{i}(\tilde{x})d\tilde{x}\}$$ is a DR estimator. It was shown in NHR (1998) that the Powell, Stock, and Stoker (1989) estimator with a twicing kernel is numerically equal to a leave one out version of this estimator for the original (before twicing) kernel. Thus the DR result for $\hat{\beta}$ gives an interpretation of the twicing kernel estimator as a DR estimator.
The expectation of the DR moment functions of both Theorem 3 and 4 are affine in $\gamma$ and $\lambda$ holding the other fixed at the truth. This property of DR moment functions is general, as we show by the following characterization of DR moment functions:
<span style="font-variant:small-caps;">Theorem 5:</span> *If* $\Gamma$ *and* $\Lambda$ *are linear then* $\psi(z,\beta,\gamma,\lambda)$ *is DR if and only if* $$\left. \partial E[\psi(z_{i},\beta_{0},(1-\tau)\gamma_{0}+\tau\gamma
,\lambda_{0})]\right\vert _{\tau=0}=0,\left. \partial E[\psi(z_{i},\beta
_{0},\gamma_{0},(1-\tau)\lambda_{0}+\tau\lambda)]\right\vert _{\tau=0}=0,$$ *and* $E[\psi(z_{i},\beta_{0},\gamma,\lambda_{0})]$ *and* $E[\psi(z_{i},\beta_{0},\gamma_{0},\lambda)]$ *are affine in* $\gamma
$ *and* $\lambda$ *respectively.*
The zero derivative condition of this result is a Gateaux derivative, componentwise version of LR. Thus, we can focus a search for DR moment conditions on those that are LR. Also, a DR moment function must have an expectation that is affine in each of $\gamma$ and $\lambda$ while the other is held fixed at the truth. It is sufficient for this condition that $\psi(z_{i},\beta_{0},\gamma,\lambda)$ be affine in each of $\gamma$ and $\lambda$ while the other is held fixed. This property can depend on how $\gamma$ and $\lambda$ are specified. For example the missing data DR moment function $m(1,u)-\beta+\pi(u)^{-1}a[y-\gamma(x)]$ is not affine in the propensity score $\pi(u)=\Pr(a_{i}=1|u_{i}=u)$ but is in $\lambda
(x)=\pi(u)^{-1}a$.
In general Theorem 5 motivates the construction of DR moment functions by adding the adjustment term to obtain a LR moment function that will then be DR if it is affine in $\gamma$ and $\lambda$ separately. It is interesting to note that in the NPIV setting of Theorem 3 and the density setting of Theorem 4 that the adjustment term is always affine in $\gamma$ and $\lambda.$ It then follows from Theorem 5 that in those settings LR moment conditions are precisely those where $E[m(z_{i},\beta_{0},\gamma)]$ is affine in $\gamma.$ Robins and Rotnitzky (2001) gave conditions for the existence of DR moment conditions in semiparametric models. Theorem 5 is complementary to those results in giving a complete characterization of DR moments when $\Gamma$ and $\Lambda$ are linear.
Assumptions 2 and 3 both specify that $E[m(z_{i},\beta_{0},\gamma)]$ is continuous in an integrated squared deviation norm. These continuity conditions are linked to finiteness of the semiparametric variance bound for the functional $E[m(z_{i},\beta_{0},\gamma)],$ as discussed in Newey and McFadden (1994) for Assumption 2 with $w_{i}=x_{i}$ and for Assumption 3. For Assumption 2 with $w_{i}\neq x_{i}$ Severini and Tripathi (2012) showed for $m(z,\beta,\gamma)=v(w)\gamma(w)-\beta$ with known $v(w)$ that the existence of $\lambda_{0}(w)$ with $v(w_{i})=E[\lambda_{0}(x_{i})|w_{i}]$ is necessary for the existence of a root-n consistent estimator of $\beta$. Thus the conditions of Assumption 2 are also linked to necessary conditions for root-n consistent estimation when $w_{i}\neq x_{i}.$
Partial robustness refers to settings where $E[m(z_{i},\beta_{0},\bar{\gamma
})]=0$ for some $\bar{\gamma}\neq\gamma_{0}$. The novel DR moment conditions given here lead to novel partial robustness results as we now demonstrate in the conditional moment restriction setting of Assumption 2. When $\lambda
_{0}(x)$ in Assumption 2 is restricted in some way there may exist $\tilde{\gamma}\neq\gamma_{0}$ with $E[\lambda_{0}(x_{i})\{y_{i}-\tilde
{\gamma}(w_{i})\}]=0.$ Then$$E[m(z_{i},\beta_{0},\tilde{\gamma})]=-E[\lambda_{0}(x_{i})\{y_{i}-\tilde{\gamma}(w_{i})\}]=0.$$ Consider the average derivative $\beta_{0}=E[\partial\gamma_{0}(w_{i})/\partial w_{r}]$ where $m(z,\beta,\gamma)=\partial\gamma(w)/\partial
w_{r}-\beta$ for some $r.$ Let $\delta=(E[a(x_{i})p(w_{i})^{\prime}])^{-1}E[a(x_{i})y_{i}]$ be the limit of the linear IV estimator with right hand side variables $p(w)$ and the same number of instruments $a(x).$ The following is a partial robustness result that provides conditions for the average derivative of the linear IV estimator to equal the true average derivative:
<span style="font-variant:small-caps;">Theorem 6:</span> If $-\partial\ln f_{0}(w)/\partial w_{r}=c^{\prime}p(w)$ for a constant vector $c$, $E[p(w_{i})p(w_{i})^{\prime}]$ is nonsingular, and $E[a(x_{i})|w_{i}=w]=\Pi p(w)$ for a square nonsingular $\Pi$ then for $\delta=(E[a(x_{i})p(w_{i})^{\prime}])^{-1}E[a(x_{i})y_{i}],$$$E[\partial\{p(w_{i})^{\prime}\delta\}/\partial w_{r}]=E[\partial\gamma
_{0}(w_{i})/\partial w_{r}].$$
This result shows that if the density score is a linear combination of the right-hand side variables $p(w)$ used by linear IV, the conditional expectation of the instruments $a(x_{i})$ given $w_{i}$ is a nonsingular linear combination of $p(w)$, and $p(w)$ has a nonsingular second moment matrix then the average derivative of the linear IV estimator is the true average derivative. This is a generalization to NPIV of Stoker’s (1986) result that linear regression coefficients equal the average derivatives when the regressors are multivariate Gaussian.
DR moment conditions can be used to identify parameters of interest. Under Assumption 1 $\beta_{0}$ may be identified from$$E[m(z_{i},\beta_{0},\bar{\gamma})]=-E[\phi(z_{i},\beta_{0},\bar{\gamma
},\lambda_{0})]$$ for any fixed $\bar{\gamma}$ when the solution $\beta_{0}$ to this equation is unique.
<span style="font-variant:small-caps;">Theorem 7:</span> *If Assumption 1 is satisfied,* $\lambda_{0}$ *is identified, and for some* $\bar{\gamma}$ *the equation* $E[\psi(z_{i},\beta,\bar{\gamma},\lambda_{0})]=0$ *has a unique solution then* $\beta_{0}$ *is identified as that solution.*
Applying this result to the NPIV setting of Assumption 2 gives an explicit formula for certain functionals of $\gamma_{0}(w)$ without requiring that the completeness identification condition of Newey and Powell (1989, 2003) be satisfied, similarly to Santos (2011). Suppose that $v(w)$ is identified, e.g. as for the weighted average derivative. Since both $w$ and $x$ are observed it follows that a solution $\lambda_{0}(x)$ to $v(w)=E[\lambda_{0}(x)|w]$ will be identified if such a solution exists. Plugging in $\bar{\gamma}=0$ into the equation $E[\psi(z_{i},\beta_{0},\bar{\gamma},\lambda_{0})]=0$ gives
<span style="font-variant:small-caps;">Corollary 8:</span> *If* $v(w_{i})$ *is identified and there exists* $\lambda_{0}(x_{i})$ *such that* $v(w_{i})=E[\lambda_{0}(x_{i})|w_{i}]$ *then* $\beta_{0}=E[v(w_{i})\gamma_{0}(w_{i})]$ *is identified as* $\beta_{0}=E[\lambda_{0}(x_{i})y_{i}]$*.*
Note that this result holds without the completeness condition. Identification of $\beta_{0}=E[v(w_{i})\gamma_{0}(w_{i})]$ for known $v(w_{i})$ with $v(w_{i})=E[\lambda_{0}(x_{i})|w_{i}]$ follows from Severini and Tripathi (2006). Corollary 8 extends that analysis to the case where $v(w_{i})$ is only identified but not necessarily known and links it to DR moment conditions. Santos (2011) gives a related formula for a parameter $\beta_{0}=\int\tilde
{v}(w)\lambda_{0}(w)dw$. The formula here differs from Santos (2011) in being an expectation rather than a Lebesgue integral. Santos (2011) constructed an estimator. That is beyond the scope of this paper.
Conditional Moment Restrictions
===============================
Models of conditional moment restrictions that depend on unknown functions are important in econometrics. In such models the nonparametric components may be determined simultaneously with the parametric components. In this setting it is useful to work directly with the instrumental variables to obtain LR moment conditions rather than to make a first step influence adjustment. For that reason we focus in this Section on constructing LR moments by orthogonalizing the instrumental variables.
Our orthogonal instruments framework is based on based on conditional moment restrictions of the form$$E[\rho_{j}(z_{i},\beta_{0},\gamma_{0})|x_{ji}]=0,(j=1,...,J),
\label{cond mom restrict}$$ where each $\rho_{j}(z,\beta,\gamma)$ is a scalar residual and $x_{j}$ are instruments that may differ across $j$. This model is considered by Chamberlain (1992) and Ai and Chen (2003, 2007) when $x_{j}$ is the same for each $j$ and for Ai and Chen (2012) when the set of $x_{j}$ includes $x_{j-1}.$ We allow the residual vector $\rho(z,\beta,\gamma)$ to depend on the entire function $\gamma$ and not just on its value at some function of the observed data $z_{i}$.
In this framework we consider LR moment functions having the form$$\psi(z,\beta,\gamma,\lambda)=\lambda(x)\rho(z,\beta,\gamma), \label{gcm}$$ where $\lambda(x)=[\lambda_{1}(x_{1}),...,\lambda_{J}(x_{J})]$ is a matrix of instrumental variables with the $j^{th}$ column given by $\lambda_{j}(x_{j}).$ We will define orthogonal instruments to be those that make $\psi
(z,\beta,\gamma,\lambda)$ locally robust. To define orthogonal instrumental variables we assume that $\gamma$ is allowed to vary over a linear set $\Gamma$ as $F$ varies. For each $\Delta\in\Gamma$ let$$\bar{\rho}_{\gamma}(x,\Delta)=(\frac{\partial E[\rho_{1}(z_{i},\beta
_{0},\gamma_{0}+\tau\Delta)|x_{1}]}{\partial\tau},...,\frac{\partial
E[\rho_{J}(z_{i},\beta_{0},\gamma_{0}+\tau\Delta)|x_{J}]}{\partial\tau
})^{\prime}.$$ This $\bar{\rho}_{\gamma}(x,\Delta)$ is the Gateaux derivative with respect to $\gamma$ of the conditional expectation of the residuals in the direction $\Delta.$ We characterize $\lambda_{0}(x)$ as orthogonal if$$E[\lambda_{0}(x_{i})\bar{\rho}_{\gamma}(x_{i},\Delta)]=0\text{ for all }\Delta\in\Gamma.$$ We assume that $\bar{\rho}_{\gamma}(x,\Delta)$ is linear in $\Delta$ and consider the Hilbert space of vectors of random vectors $a(x)=$ $(a_{1}(x_{1}),...,a_{J}(x_{J}))$ with inner product $\left\langle a,b\right\rangle
=E[a(x_{i})^{\prime}b(x_{i})]$. Let $\bar{\Lambda}_{\gamma}$ denote the closure of the set $\{\bar{\rho}_{\gamma}(x,\Delta):\Delta\in\Gamma\}$ in that Hilbert space. Orthogonal instruments are those where each row of $\lambda
_{0}(x)$ is orthogonal to $\bar{\Lambda}_{\gamma}.$ They can be interpreted as instrumental variables where the effect of estimation of $\gamma$ has been partialed out. When $\lambda_{0}(x)$ is orthogonal then $\psi(z,\beta
,\gamma,\lambda)=\lambda(x)\rho(z,\beta,\gamma)$ is LR:
<span style="font-variant:small-caps;">Theorem 9:</span> *If each row of* $\lambda_{0}(x)$ *is orthogonal to* $\bar{\Lambda}_{\gamma}$ *then the moment functions in equation (\[gcm\]) are LR.*
We also have a DR result:
<span style="font-variant:small-caps;">Theorem 10:</span> *If each row of* $\lambda_{0}(x)$ *is orthogonal to* $\bar{\Lambda}_{\gamma}$ *and* $\rho(z,\beta,\gamma
)$ *is affine in* $\gamma\in\Gamma$ *then the moment functions in equation (\[gcm\]) are DR for* $\Lambda=\{\lambda(x):$ ** $E[\lambda(x_{i})^{\prime}\rho(z_{i},\beta_{0},\gamma_{0})^{\prime}\rho
(z_{i},\beta_{0},\gamma_{0})\lambda(x_{i})]$.
There are many ways to construct orthogonal instruments. For instance, given a $r\times(J-1)$ matrix of instrumental variables $\lambda(x)$ one could construct corresponding orthogonal ones $\lambda_{0}(x_{i})$ as the matrix where each row of $\lambda(x)$ is replaced by the residual from the least squares projection of the corresponding row of $\lambda(x)$ on $\bar{\Lambda
}_{\gamma}$. For local identification of $\beta$ we also require that $$rank(\left. \partial E[\psi(z_{i},\beta,\gamma_{0})]/\partial\beta\right\vert
_{\beta=\beta_{0}})=\dim(\beta). \label{local id beta}$$
A model where $\beta_{0}$ is identified from semiparametric conditional moment restrictions with common instrumental variables is a special case where $x_{ji}$ is the same for each $j$. In this case there is a way to construct orthogonal instruments that leads to an efficient estimator of $\beta_{0}$. Let $\Sigma(x_{i})$ denote some positive definite matrix with its smallest eigenvalue bounded away from zero, so that $\Sigma(x_{i})^{-1}$ is bounded. Let $\left\langle a,b\right\rangle _{\Sigma}=E[a(x_{i})^{\prime}\Sigma
(x_{i})^{-1}b(x_{i})]$ denote an inner product and note that $\bar{\Lambda
}_{\gamma}$ is closed in this inner product by $\Sigma(x_{i})^{-1}$ bounded. Let $\tilde{\lambda}_{k}^{\Sigma}(x_{i},\lambda)$ denote the residual from the least squares projection of the $k^{th}$ row $\lambda\left( x\right)
^{\prime}e_{k}$ of $\lambda(x)$ on $\bar{\Lambda}_{\gamma}$ with the inner product $\left\langle a,b\right\rangle _{\Sigma}.$ Then for all $\Delta
\in\Gamma,$ $$E[\tilde{\lambda}_{k}^{\Sigma}(x_{i},\lambda)^{\prime}\Sigma(x_{i})^{-1}\bar{\rho}_{\gamma}(x_{i},\Delta)]=0,$$ so that for $\tilde{\lambda}^{\Sigma}(x_{i},\lambda)=[\tilde{\lambda}_{1}^{\Sigma}(x_{i},\lambda),...,\tilde{\lambda}_{r}^{\Sigma}(x_{i},\lambda)]$ the instrumental variables $\tilde{\lambda}^{\Sigma}(x_{i},\lambda
)\Sigma(x_{i})^{-1}$ are orthogonal. Also, $\tilde{\lambda}^{\Sigma}(x_{i},\lambda)$ can be interpreted as the solution to$$\min_{\{D(x):D(x)^{\prime}e_{k}\in\bar{\Lambda}_{\gamma},k=1,...,r\}}tr(E[\{\lambda(x_{i})-D(x_{i})\}\Sigma(x_{i})^{-1}\{\lambda(x_{i})-D(x_{i})\}^{\prime}])$$ where the minimization is in the positive semidefinite sense.
The orthogonal instruments that minimize the asymptotic variance of GMM in the class of GMM estimators with orthogonal instruments are given by$$\lambda_{0}^{\ast}(x)=\tilde{\lambda}^{\Sigma^{\ast}}(x,\lambda_{\beta})\Sigma^{\ast}(x)^{-1},\lambda_{\beta}(x_{i})=\left. \frac{\partial
E[\rho(z_{i},\beta,\gamma_{0})|x_{i}]}{\partial\beta}\right\vert _{\beta
=\beta_{0}}^{\prime},\Sigma^{\ast}(x_{i})=Var(\rho_{i}|x_{i}),\rho_{i}=\rho(z_{i},\beta_{0},\gamma_{0}).$$
<span style="font-variant:small-caps;">Theorem 11:</span> *The instruments* $\varphi^{\ast}(x_{i})$ *give an efficient estimator in the class of IV estimators with orthogonal instruments.*
The asymptotic variance of the GMM estimator with optimal orthogonal instruments is $$(E[m_{i}^{\ast}m_{i}^{\ast\prime}])^{-1}=E[\tilde{\lambda}(x_{i},\lambda
^{\ast},\Sigma^{\ast})\Sigma^{\ast}(x_{i})^{-1}\tilde{\lambda}(x_{i},\lambda^{\ast},\Sigma^{\ast})^{\prime}])^{-1}.$$ This matrix coincides with the semiparametric variance bound of Ai and Chen (2003). Estimation of the optimal orthogonal instruments is beyond the scope of this paper. The series estimator of Ai and Chen (2003) could be used for this.
This framework includes moment restrictions with a NPIV first step $\gamma$ satisfying $E[\rho(z_{i},\gamma_{0})|x_{i}]=0$ where we can specify $\rho
_{1}(z,\beta,\gamma)=m(z,\beta,\gamma),$ $x_{1i}=1,$ $\rho_{2}(z,\beta
,\gamma)=\rho(z,\gamma),$ and $x_{2i}=x_{i}.$ It generalizes that setup by allowing for more residuals $\rho_{j}(z,\beta,\gamma)$, $(j\geq3)$ and allowing all residuals to depend on $\beta.$
Asymptotic Theory
=================
In this Section we give simple and general asymptotic theory for LR estimators that incorporates the cross-fitting of equation (\[cfit\]). Throughout we use the structure of LR moment functions that are the sum $\psi(z,\beta
,\gamma,\lambda)=m(z,\beta,\gamma)+\phi(z,\beta,\gamma,\lambda)$ of an identifying or original moment function $m(z,\beta,\gamma)$ depending on a first step function $\gamma$ and an influence adjustment term $\phi
(z,\beta,\gamma,\lambda)$ that can depend on an additional first step $\lambda.$ The asymptotic theory will apply to any moment function that can be decomposed into a function of a single nonparametric estimator and a function of two nonparametric estimators. This structure and LR leads to particularly simple and general conditions.
The conditions we give are composed of mean square consistency conditions for first steps and one, two, or three rate conditions for quadratic remainders. We will only use one quadratic remainder rate for DR moment conditions, involving faster than $1/\sqrt{n}$ convergence of products of estimation errors for $\hat{\gamma}$ and $\hat{\lambda}.$ When $E[m(z_{i},\beta
_{0},\gamma)+\phi(z_{i},\beta_{0},\gamma,\lambda_{0})]$ is not affine in $\gamma$ we will impose a second rate condition that involves faster than $n^{-1/4}$ convergence of $\hat{\gamma}.$ When $E[\phi(z_{i},\gamma
_{0},\lambda)]$ is also not affine in $\lambda$ we will impose a third rate condition that involves faster than $n^{-1/4}$ convergence of $\hat{\lambda}.$ Most adjustment terms $\phi(z,\beta,\gamma,\lambda)$ of which we are aware, including for first step conditional moment restrictions and densities, have $E[\phi(z_{i},\beta_{0},\gamma_{0},\lambda)]$ affine in $\lambda,$ so that faster $n^{-1/4}$ convergence of $\hat{\lambda}$ will not be required under our conditions. It will suffice for most LR estimators which we know of to have faster than $n^{-1/4}$ convergence of $\hat{\gamma}$ and faster than $1/\sqrt{n}$ convergence of the product of estimation errors for $\hat{\gamma
}$ and $\hat{\lambda},$ with only the latter condition imposed for DR moment functions. We also impose some additional conditions for convergence of the Jacobian of the moments and sample second moments that give asymptotic normality and consistent asymptotic variance estimation for $\hat{\beta}$.
An important intermediate result for asymptotic normality is$$\sqrt{n}\hat{\psi}(\beta_{0})=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\psi(z_{i},\beta_{0},\gamma_{0},\lambda_{0})+o_{p}(1), \label{no effec}$$ where $\hat{\psi}(\beta)$ is the cross-fit, sample, LR moments of equation (\[cfit\]). This result will mean that the presence of the first step estimators has no effect on the limiting distribution of the moments at the true $\beta_{0}$. To formulate conditions for this result we decompose the difference between the left and right-hand sides into several remainders. Let $\phi(z,\gamma,\lambda)=\phi(z,\beta_{0},\gamma,\lambda),$ $\bar{\phi}(\gamma,\lambda)=E[\phi(z_{i},\gamma,\lambda)],$ and $\bar{m}(\gamma
)=E[m(z_{i},\beta_{0},\gamma)],$ so that $\bar{\psi}(\gamma,\lambda)=\bar
{m}(\gamma)+\bar{\phi}(\gamma,\lambda)$ Then adding and subtracting terms gives $$\sqrt{n}[\hat{\psi}(\beta_{0})-\sum_{i=1}^{n}\psi(z_{i},\beta_{0},\gamma
_{0},\lambda_{0})/n]=\hat{R}_{1}+\hat{R}_{2}+\hat{R}_{3}+\hat{R}_{4},
\label{redecomp}$$ where$$\begin{aligned}
\hat{R}_{1} & =\frac{1}{\sqrt{n}}\sum_{i=1}^{n}[m(z_{i},\beta_{0},\hat{\gamma}_{i})-m(z_{i},\beta_{0},\gamma_{0})-\bar{m}(\hat{\gamma}_{i})]\label{remain}\\
& +\frac{1}{\sqrt{n}}\sum_{i=1}^{n}[\phi(z_{i},\hat{\gamma}_{i},\lambda
_{0})-\phi(z_{i},\gamma_{0},\lambda_{0})-\bar{\phi}(\hat{\gamma}_{i},\lambda_{0})+\phi(z_{i},\gamma_{0},\hat{\lambda}_{i})-\phi(z_{i},\gamma
_{0},\lambda_{0})-\bar{\phi}(\gamma_{0},\hat{\lambda}_{i})],\nonumber\\
\hat{R}_{2} & =\frac{1}{\sqrt{n}}\sum_{i=1}^{n}[\phi(z_{i},\hat{\gamma}_{i},\hat{\lambda}_{i})-\phi(z_{i},\hat{\gamma}_{i},\lambda_{0})-\phi
(z_{i},\gamma_{0},\hat{\lambda}_{i})+\phi(z_{i},\gamma_{0},\lambda
_{0})],\nonumber\\
\hat{R}_{3} & =\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\bar{\psi}(\hat{\gamma}_{i},\lambda_{0}),\;\;\;\hat{R}_{4}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\bar{\phi
}(\gamma_{0},\hat{\lambda}_{i}),\nonumber\end{aligned}$$
We specify regularity conditions sufficient for each of $\hat{R}_{1}$, $\hat{R}_{2}$, $\hat{R}_{3},$ and $\hat{R}_{4}$ to converge in probability to zero so that equation (\[no effec\]) will hold. The remainder term $\hat
{R}_{1}$ is a stochastic equicontinuity term as in Andrews (1994). We give mean square consistency conditions for $\hat{R}_{1}\overset{p}{\longrightarrow
}0$ in Assumption 3.
The remainder term $\hat{R}_{2}$ is a second order remainder that involves both $\hat{\gamma}$ and $\hat{\lambda}.$ When the influence adjustment is $\phi(z,\gamma,\lambda)=\lambda(x)[y-\gamma(w)],$ as for conditional moment restrictions, then$$\hat{R}_{2}=\frac{-1}{\sqrt{n}}\sum_{i=1}^{n}[\hat{\lambda}_{i}(x_{i})-\lambda_{0}(x_{i})][\hat{\gamma}_{i}(w_{i})-\gamma_{0}(w_{i})].$$ $\hat{R}_{2}$ will converge to zero when the product of convergence rates for $\hat{\lambda}_{i}(x_{i})$ and $\hat{\gamma}_{i}(w_{i})$ is faster than $1/\sqrt{n}.$ However, that is not the weakest possible condition. Weaker conditions for locally linear regression first steps are given by Firpo and Rothe (2017) and for series regression first steps by Newey and Robins (2017). These weaker conditions still require that the product of biases of $\hat{\lambda}_{i}(x_{i})$ and $\hat{\gamma}_{i}(w_{i})$ converge to zero faster than $1/\sqrt{n}$ but have weaker conditions for variance terms. We allow for these weaker conditions by allowing $\hat{R}_{2}\overset{p}{\longrightarrow}0$ as a regularity condition. Assumption 5 gives these conditions.
We will have $\hat{R}_{3}=\hat{R}_{4}=0$ in the DR case of Assumption 1, where $\hat{R}_{1}\overset{p}{\longrightarrow}0$ and $\hat{R}_{2}\overset{p}{\longrightarrow}0$ will suffice for equation (\[no effec\]). In non DR cases LR leads to $\bar{\psi}(\gamma,\lambda_{0})=\bar{m}(\gamma
)+\bar{\phi}(\gamma,\lambda_{0})$ having a zero functional derivative with respect to $\gamma$ at $\gamma_{0}$ so that $\hat{R}_{3}\overset{p}{\longrightarrow}0$ when $\hat{\gamma}_{i}$ converges to $\gamma_{0}$ at a rapid enough, feasible rate. For example if $\bar{\psi
}(\gamma,\lambda_{0})$ is twice continuously Frechet differentiable in a neighborhood of $\gamma_{0}$ for a norm $\left\Vert \cdot\right\Vert ,$ with zero Frechet derivative at $\gamma_{0}$. Then$$\left\vert \hat{R}_{3}\right\vert \leq C\sum_{\ell=1}^{L}\sqrt{n}\left\Vert
\hat{\gamma}_{\ell}-\gamma_{0}\right\Vert ^{2}\overset{p}{\longrightarrow}0$$ when $\left\Vert \hat{\gamma}-\gamma_{0}\right\Vert =o_{p}(n^{-1/4})$. Here $\hat{R}_{3}\overset{p}{\longrightarrow}0$ when each $\hat{\gamma}_{\ell}$ converges to $\gamma_{0}$ more quickly than $n^{-1/4}$. It may be possible to weaken this condition by bias correcting $m(z,\beta,\hat{\gamma}),$ as by the bootstrap in Cattaneo and Jansson (2017), by the jackknife in Cattaneo Ma and Jansson (2017), and by cross-fitting in Newey and Robins (2017). Consideration of such bias corrections for $m(z,\beta,\hat{\gamma})$ is beyond the scope of this paper.
In many cases $\hat{R}_{4}=0$ even though the moment conditions are not DR. For example that is true when $\hat{\gamma}$ is a pdf or when $\gamma_{0}$ estimates the solution to a conditional moment restriction. In such cases mean square consistency, $\hat{R}_{2}\overset{p}{\longrightarrow}0,$ and faster than $n^{-1/4}$ consistency of $\hat{\gamma}$ suffices for equation (\[no effec\]); no convergence rate for $\hat{\lambda}$ is needed. The simplification that $\hat{R}_{4}=0$ seems to be the result of $\lambda$ being a Riesz representer for the linear functional that is the derivative of $\bar{m}(\gamma)$ with respect to $\gamma.$ Such a Riesz representer will enter $\bar{\phi}(\lambda,\gamma_{0})$ linearly, leading to $\hat{R}_{4}=0.$ When $\hat{R}_{4}\neq0$ then $\hat{R}_{4}\overset{p}{\longrightarrow}0$ will follow from twice Frechet differentiability of $\bar{\phi}(\lambda,\gamma
_{0})$ in $\lambda$ and faster than $n^{-1/4}$ convergence of $\hat{\lambda}.$
All of the conditions can be easily checked for a wide variety of machine learning and conventional nonparametric estimators. There are well known conditions for mean square consistency for many conventional and machine learning methods. Rates for products of estimation errors are also know for many first step estimators as are conditions for $n^{-1/4}$ consistency. Thus, the simple conditions we give here are general enough to apply to a wide variety of first step estimators.
The first formal assumption of this section is sufficient for $\hat{R}_{1}\overset{p}{\longrightarrow}0.$
<span style="font-variant:small-caps;">Assumption 4:</span> *For each* $\ell=1,...,L$*, i) Either* $m(z,\beta_{0},\gamma)$ *does not depend on* $z$ *or* $\int\{m(z,\beta_{0},\hat{\gamma}_{\ell})-m(z,\beta_{0},\gamma_{0})\}^{2}F_{0}(dz)\overset{p}{\longrightarrow}0,$ *ii)* $\int\{\phi
(z,\hat{\gamma}_{\ell},\lambda_{0})-\phi(z,\gamma_{0},\lambda_{0})\}^{2}F_{0}(dz)\overset{p}{\longrightarrow}0,$ *and* $\int\{\phi(z,\gamma
_{0},\hat{\lambda}_{\ell})-\phi(z,\gamma_{0},\lambda_{0})\}^{2}F_{0}(dz)\overset{p}{\longrightarrow}0;$
The cross-fitting used in the construction of $\hat{\psi}(\beta_{0})$ is what makes the mean-square consistency conditions of Assumption 4 sufficient for $\hat{R}_{1}\overset{p}{\longrightarrow}0$. The next condition is sufficient for $\hat{R}_{2}\overset{p}{\longrightarrow}0.$
<span style="font-variant:small-caps;">Assumption 5:</span> *For each* $\ell=1,...,L$*, either i)*$$\sqrt{n}\int\max_{j}|\phi_{j}(z,\hat{\gamma}_{\ell},\hat{\lambda}_{\ell})-\phi_{j}(z,\gamma_{0},\hat{\lambda}_{\ell})-\phi_{j}(z,\hat{\gamma}_{\ell
},\lambda_{0})+\phi_{j}(z,\gamma_{0},\lambda_{0})|F_{0}(dz)\overset{p}{\longrightarrow}0$$ *or ii)* $\hat{R}_{2}\overset{p}{\longrightarrow}0.$
As previously discussed, this condition allows for just $\hat{R}_{2}\overset{p}{\longrightarrow}0$ in order to allow the weak regularity conditions of Firpo and Rothe (2017) and Newey and Robins (2017). The first result of this Section shows that Assumptions 4 and 5 are sufficient for equation (*\[no effec\]*) when the moment functions are DR.
<span style="font-variant:small-caps;">Lemma 12:</span> *If Assumption 1 is satisfied, with probability approaching one* $\hat{\gamma}\in\Gamma$*,* $\hat{\lambda}\in\Lambda
,$ *and Assumptions 4 and 5 are satisfied then equation (\[no effec\]) is satisfied.*
An important class of DR estimators are those from equation (\[drlin\]). The following result gives conditions for asymptotic linearity of these estimators:
<span style="font-variant:small-caps;">Theorem 13:</span> *If a) Assumptions 2 and 4 i) are satisfied with* $\hat{\gamma}\in\Gamma$ *and* $\hat{\lambda}\in\Lambda$ *with probability approaching one; b)* $\lambda_{0}(x_{i})$ *and* $E[\{y_{i}-\gamma_{0}(w_{i})\}^{2}|x_{i}]$ *are bounded; c) for each* $\ell=1,...,L$*,* $\int[\hat{\gamma}_{\ell}(w)-\gamma_{0}(w)]^{2}F_{0}(dz)\overset{p}{\longrightarrow}0,$ ** $\int[\hat{\lambda}_{\ell
}(x)-\lambda_{0}(x)]^{2}F_{0}(dz)$ ** $\overset{p}{\longrightarrow}0$*, and either*$$\sqrt{n}\left\{ \int[\hat{\gamma}_{\ell}(w)-\gamma_{0}(w)]^{2}F_{0}(dw)\right\} ^{1/2}\left\{ \int[\hat{\lambda}_{\ell}(x)-\lambda_{0}(x)]^{2}F_{0}(dx)\right\} ^{1/2}\mathit{\ }\overset{p}{\longrightarrow}0$$ *or*$$\frac{1}{\sqrt{n}}\sum_{i\in I_{\ell}}\{\hat{\gamma}_{\ell}(w_{i})-\gamma
_{0}(w_{i})\}\{\hat{\lambda}_{\ell}(x_{i})-\lambda_{0}(x_{i})\}\overset{p}{\longrightarrow}0;$$ *then*$$\sqrt{n}(\hat{\beta}-\beta_{0})=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}[g(z_{i},\gamma_{0})-\beta_{0}+\lambda_{0}(x_{i})\{y_{i}-\gamma_{0}(w_{i})\}]+o_{p}(1).$$
The conditions of this result are simple, general, and allow for machine learning first steps. Conditions a) and b) simply require mean square consistency of the first step estimators $\hat{\gamma}$ and $\hat{\lambda}.$ The only convergence rate condition is c), which requires a product of estimation errors for the two first steps to go to zero faster than $1/\sqrt{n}$. This condition allows for a trade-off in convergence rates between the two first steps, and can be satisfied even when one of the two rates is not very fast. This trade-off can be important when $\lambda_{0}(x)$ is not continuous in one of the components of $x$, as in the surplus bound example. Discontinuity in $x$ can limit that rate at which $\lambda_{0}(x)$ can be estimated. This result extends the results of Chernozhukov et al. (2018) and Farrell (2015) for DR estimators of treatment effects to the whole novel class of DR estimators from equation (\[drlin\]) with machine learning first steps. In interesting related work, Athey et al. (2016) show root-n consistent estimation of an average treatment effect is possible under very weak conditions on the propensity score, under strong sparsity of the regression function. Thus, for machine learning the conditions here and in Athey et al. (2016) are complementary and one may prefer either depending on whether or not the regression function can be estimated extremely well based on a sparse method. The results here apply to many more DR moment conditions.
DR moment conditions have the special feature that $\hat{R}_{3}$ and $\hat
{R}_{4}$ in Proposition 4 are equal to zero. For estimators that are not DR we impose that $\hat{R}_{3}$ and $\hat{R}_{4}$ converge to zero.
<span style="font-variant:small-caps;">Assumption 6:</span> *For each* $\ell=1,...,L$*, i)* $\sqrt
{n}\bar{\psi}(\hat{\gamma}_{\ell},\lambda_{0})\overset{p}{\longrightarrow}0$ *and ii)* $\sqrt{n}\bar{\phi}(\gamma_{0},\hat{\lambda}_{\ell
})\overset{p}{\longrightarrow}0.$
Assumption 6 requires that $\hat{\gamma}$ converge to $\gamma_{0}$ rapidly enough but places no restrictions on the convergence rate of $\hat{\lambda}$ when $\bar{\phi}(\gamma_{0},\hat{\lambda}_{\ell})=0.$
<span style="font-variant:small-caps;">Lemma 14:</span> *If Assumptions 4-6 are satisfied then equation (\[no effec\]) is satisfied.*
Assumptions 4-6 are based on the decomposition of LR moment functions into an identifying part and an influence function adjustment. These conditions differ from other previous work in semiparametric estimation, as in Andrews (1994), Newey (1994), Newey and McFadden (1994), Chen, Linton, and van Keilegom (2003), Ichimura and Lee (2010), Escanciano et al. (2016), and Chernozhukov et al. (2018), that are not based on this decomposition. The conditions extend Chernozhukov et. al. (2018) to many more DR estimators and to estimators that are nonlinear in $\hat{\gamma}$ but only require a convergence rate for $\hat{\gamma}$ and not for $\hat{\lambda}$.
This framework helps explain the potential problems with “plugging in” a first step machine learning estimator into a moment function that is not LR. Lemma 14 implies that if Assumptions 4-6 are satisfied for some $\hat{\lambda}$ then $\sqrt{n}\hat{m}(\beta_{0})-\sum_{i=1}^{n}\psi(z_{i},\beta_{0},\gamma
_{0})/\sqrt{n}\overset{p}{\longrightarrow}0$ if and only if$$\hat{R}_{5}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\phi(z_{i},\hat{\gamma},\hat{\lambda})\overset{p}{\longrightarrow}0. \label{plugin}$$ The plug-in method will fail when this equation does not hold. For example, suppose $\gamma_{0}=E[y|x]$ so that by Proposition 4 of Newey (1994),$$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\phi(z_{i},\hat{\gamma},\hat{\lambda})=\frac{-1}{\sqrt{n}}\sum_{i=1}^{n}\hat{\lambda}_{i}(x_{i})[y_{i}-\hat{\gamma
}_{i}(x_{i})].$$ Here $\hat{R}_{5}\overset{p}{\longrightarrow}0$ is an approximate orthogonality condition between the approximation $\hat{\lambda}_{i}(x_{i})$ to $\lambda_{0}(x_{i})$ and the nonparametric first stage residuals $y_{i}-\hat{\gamma}_{i}(x_{i}).$ Machine learning uses model selection in the construction of $\hat{\gamma}_{i}(x_{i}).$ If the model selected by $\hat{\gamma}_{i}(x_{i})$ to approximate $\gamma_{0}(x_{i})$ is not rich (or dense) enough to also approximate $\lambda_{0}(x_{i})$ then $\hat{\lambda}_{i}(x_{i})$ need not be approximately orthogonal to $y_{i}-\hat{\gamma}_{i}(x_{i})$ and $\hat{R}_{5}$ need not converge to zero. In particular, if the variables selected to be used to approximate $\gamma_{0}(x_{i})$ cannot be used to also approximate $\lambda_{0}(x_{i})$ then the approximate orthogonality condition can fail. This phenomenon helps explain the poor performance of the plug-in estimator shown in Belloni, Chernozhukov, and Hansen (2014) and Chernozhukov et al. (2017, 2018). The plug-in estimator can be root-n consistent if the only thing being selected is an overall order of approximation, as in the series estimation results of Newey (1994). General conditions for root-n consistency of the plug-in estimator can be formulated using Assumptions 4-6 and $\hat{R}_{2}\overset{p}{\longrightarrow}0,$ which we do in Appendix D.
Another component of an asymptotic normality result is convergence of the Jacobian term $\partial\hat{\psi}(\beta)/\partial\beta$ to $M=\left.
E[\partial\psi(z_{i},\beta,\gamma_{0},\lambda_{0})/\partial\beta\right\vert
_{\beta=\beta_{0}}].$ We impose the following condition for this purpose.
<span style="font-variant:small-caps;">Assumption 7:</span> $M\,$*exists and there is a neighborhood* $\mathcal{N}$ *of* $\beta_{0}$ *and* $\left\Vert \cdot
\right\Vert $ *such that i) for each* $\ell,$ $\left\Vert \hat{\gamma
}_{\ell}-\gamma_{0}\right\Vert \overset{p}{\longrightarrow}0,$ $\left\Vert
\hat{\lambda}_{\ell}-\lambda_{0}\right\Vert \overset{p}{\longrightarrow}0;$ *ii)* for all $\left\Vert \gamma-\gamma_{0}\right\Vert $ and $\left\Vert \lambda-\lambda_{0}\right\Vert $ small enough $\psi(z_{i},\beta,\gamma,\lambda)$ *is differentiable in* $\beta$ *on* $\mathcal{N}$ *with probability approaching* $1$ *iii) there is* $\zeta^{\prime}>0$ *and* $d(z_{i})$ *with* $E[d(z_{i})]<\infty
$ *such that for* $\beta\in N$ *and* $\left\Vert \gamma
-\gamma_{0}\right\Vert $ *small enough* $$\left\Vert \frac{\partial\psi(z_{i},\beta,\gamma,\lambda)}{\partial\beta
}-\frac{\partial\psi(z_{i},\beta_{0},\gamma,\lambda)}{\partial\beta
}\right\Vert \leq d(z_{i})\left\Vert \beta-\beta_{0}\right\Vert ^{\zeta
^{\prime}};$$ *iii) For each* $\ell=1,...,L,$ $j,$ and $k$, $\int\left\vert
\partial\psi_{j}(z,\beta_{0},\hat{\gamma}_{\ell},\hat{\lambda}_{\ell
})/\partial\beta_{k}-\partial\psi_{j}(z,\beta_{0},\gamma_{0},\lambda
_{0})/\partial\beta_{k}\right\vert F_{0}(dz)\overset{p}{\longrightarrow}0,$
The following intermediate result gives Jacobian convergence.
<span style="font-variant:small-caps;">Lemma 15:</span> *If Assumption 7 is satisfied then for any* $\bar{\beta}\overset{p}{\longrightarrow}\beta_{0},$ ** $\hat{\psi}(\beta)$ *is differentiable at* $\bar{\beta}$ *with probability approaching one and* $\partial\hat{\psi}(\bar{\beta})/\partial\beta
\overset{p}{\longrightarrow}M.$
With these results in place the asymptotic normality of semiparametric GMM follows in a standard way.
<span style="font-variant:small-caps;">Theorem 16:</span> *If Assumptions 4-7 are satisfied,* $\hat{\beta
}\overset{p}{\longrightarrow}\beta_{0},$ ** $\hat{W}\overset{p}{\longrightarrow}W$*,* $M^{\prime}WM$ *is nonsingular, and* $E[\left\Vert \psi(z_{i},\beta_{0},\gamma_{0},\lambda
_{0})\right\Vert ^{2}]<\infty$ *then for* $\Omega=E[\psi(z_{i},\beta_{0},\gamma_{0},\lambda_{0})\psi(z_{i},\beta_{0},\gamma_{0},\lambda
_{0})^{\prime}],$$$\sqrt{n}(\hat{\beta}-\beta_{0})\overset{d}{\longrightarrow}N(0,V),V=(M^{\prime
}WM)^{-1}M^{\prime}W\Omega WM(M^{\prime}WM)^{-1}.$$
It is also useful to have a consistent estimator of the asymptotic variance of $\hat{\beta}$. As usual such an estimator can be constructed as$$\begin{aligned}
\hat{V} & =(\hat{M}^{\prime}\hat{W}\hat{M})^{-1}\hat{M}^{\prime}\hat{W}\hat{\Omega}\hat{W}\hat{M}(\hat{M}^{\prime}\hat{W}\hat{M})^{-1},\\
\hat{M} & =\frac{\partial\hat{\psi}(\hat{\beta})}{\partial\beta},\hat
{\Omega}=\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in\mathcal{I}_{\ell}}\psi
(z_{i},\hat{\beta},\hat{\gamma}_{\ell},\hat{\lambda}_{\ell})\psi(z_{i},\hat{\beta},\hat{\gamma}_{\ell},\hat{\lambda}_{\ell})^{\prime}.\end{aligned}$$ Note that this variance estimator ignores the estimation of $\gamma$ and $\lambda$ which works here because the moment conditions are LR. The following result gives conditions for consistency of $\hat{V}.$
<span style="font-variant:small-caps;">Theorem 17:</span> *If Assumptions 4 and 7 are satisfied with* $E[b(z_{i})^{2}]<\infty,$ ** $M^{\prime}WM$ *is nonsingular, and* $$\int\left\Vert \phi(z,\hat{\gamma}_{\ell},\hat{\lambda}_{\ell})-\phi
(z,\gamma_{0},\hat{\lambda}_{\ell})-\phi(z,\hat{\gamma}_{\ell},\lambda
_{0})+\phi(z,\gamma_{0},\lambda_{0})\right\Vert ^{2}F_{0}(dz)\overset{p}{\longrightarrow}0$$ *then* $\hat{\Omega}\overset{p}{\longrightarrow}\Omega$ *and* $\hat{V}\overset{p}{\longrightarrow}V.$
In this section we have used cross-fitting and a decomposition of moment conditions into identifying and influence adjustment components to formulate simple and general conditions for asymptotic normality of LR GMM estimators. For reducing higher order bias and variance it may be desirable to let the number of groups grow with the sample size. That case is beyond the scope of this paper.
Appendix A: Proofs of Theorems
==============================
**Proof of Theorem 1:** By ii) and iii), $$0=(1-\tau)\int\phi(z,F_{\tau})F_{0}(dz)+\tau\int\phi(z,F_{\tau})G(dz).$$ Dividing by $\tau$ and solving gives$$\frac{1}{\tau}\int\phi(z,F_{\tau})F_{0}(dz)=-\int\phi(z,F_{\tau})G(dz)+\int\phi(z,F_{\tau})F_{0}(z).$$ Taking limits as $\tau\longrightarrow0$, $\tau>0$ and using i) gives$$\frac{d}{d\tau}\int\phi(z,F_{\tau})F_{0}(dz)=-\int\phi(z,F_{0})G(dz)+0=-\frac
{d\mu(F_{\tau})}{d\tau}.\text{ }Q.E.D.$$
**Proof of Theorem 2**: We begin by deriving $\phi_{1},$ the adjustment term for the first step CCP estimation. We use the definitions given in the body of the paper. We also let$$\begin{aligned}
P_{\tilde{v}j}(\tilde{v}) & =\partial P(\tilde{v})/\partial\tilde{v}_{j},\text{ }\pi_{1}=\Pr(y_{t1}=1),\text{ }\lambda_{10}(x)=E[y_{1t}|x_{t+1}=x],\\
\lambda_{j0}(x) & =E[A(x_{t})P_{\tilde{v}j}(\tilde{v}_{t})\frac{y_{tj}}{P_{j}(\tilde{v}_{t})}|x_{t+1}=x],(j=2,...,J).\end{aligned}$$ Consider a parametric submodel as described in Section 4 and let $\gamma
_{1}(x,\tau)$ denote the conditional expectation of $y_{t}$ given $x_{t}$ under the parametric submodel. Note that for $\tilde{v}_{t}=\tilde{v}(x_{t}),$$$\begin{aligned}
& E[A(x_{t})P_{\tilde{v}j}(\tilde{v}_{t})\frac{\partial E[H(\gamma
_{1}(x_{t+1},\tau))|x_{t},y_{tj}=1]}{\partial\tau}]\\
& =\frac{\partial}{\partial\tau}E[A(x_{t})P_{vj}(\tilde{v}_{t})\frac{y_{tj}}{P_{j}(\tilde{v}_{t})}H(\gamma_{1}(x_{t+1},\tau))]\\
& =\frac{\partial}{\partial\tau}E[E[A(x_{t})P_{vj}(\tilde{v}_{t})\frac
{y_{tj}}{P_{j}(\tilde{v}_{t})}|x_{t+1}]H(\gamma_{1}(x_{t+1},\tau))]\\
& =\frac{\partial}{\partial\tau}E[\lambda_{j0}(x_{t+1})H(\gamma_{1}(x_{t+1},\tau))]=\frac{\partial}{\partial\tau}E[\lambda_{j0}(x_{t})H(\gamma_{1}(x_{t},\tau))]\\
& =E[\lambda_{j0}(x_{t})\frac{\partial H(\gamma_{10}(x_{t}))}{\partial
P}^{\prime}\frac{\partial\gamma_{1}(x_{t},\tau)}{\partial\tau}]=E[\lambda
_{j0}(x_{t})\frac{\partial H(\gamma_{10}(x_{t}))}{\partial P}^{\prime}\{y_{t}-\gamma_{10}(x_{t})\}S(z_{t})].\end{aligned}$$ where the last (sixth) equality follows as in Proposition 4 of Newey (1994a), and the fourth equality follows by equality of the marginal distributions of $x_{t}$ and $x_{t+1}$. Similarly, for $\pi_{1}=\Pr(y_{t1}=1)$ and $\lambda_{10}(x)=E[y_{1t}|x_{t+1}=x]$ we have$$\begin{aligned}
\frac{\partial E[H(\gamma_{1}(x_{t+1},\tau))|y_{t1}=1]}{\partial\tau} &
=\frac{\partial E[\pi_{1}^{-1}y_{1t}H(\gamma_{1}(x_{t+1},\tau))]}{\partial
\tau}=\frac{\partial E[\pi_{1}^{-1}\lambda_{10}(x_{t+1})H(\gamma_{1}(x_{t+1},\tau))]}{\partial\tau}\\
& =\frac{\partial E[\pi_{1}^{-1}\lambda_{10}(x_{t})H(\gamma_{1}(x_{t},\tau))]}{\partial\tau}\\
& =E[\pi_{1}^{-1}\lambda_{10}(x_{t})\frac{\partial H(\gamma_{10}(x_{t}))}{\partial P}^{\prime}\{y_{t}-\gamma_{10}(x_{t})\}S(z_{t})]\end{aligned}$$ Then combining terms gives$$\begin{aligned}
& \frac{\partial E[m(z_{t},\beta_{0},\gamma_{1}(\tau),\gamma_{-10})]}{\partial\tau}\\
& =-\delta\sum_{j=2}^{J}\{E[A(x_{t})P_{vj}(\tilde{v}_{t})\frac{\partial
E[H(\gamma_{1}(x_{t+1},\tau))|x_{t},y_{tj}=1]}{\partial\tau}]\\
& -E[A(x_{t})P_{vj}(\tilde{v}_{t})]\frac{\partial E[H(\gamma_{1}(x_{t+1},\tau))|y_{t1}=1]}{\partial\tau}\}\\
& =-\delta\sum_{j=2}^{J}E[\{\lambda_{j0}(x_{t})-E[A(x_{t})P_{\tilde{v}j}(\tilde{v}_{t})]\pi_{1}^{-1}\lambda_{10}(x_{t})\}\frac{\partial
H(\gamma_{10}(x_{t}))}{\partial P}^{\prime}\{y_{t}-\gamma_{10}(x_{t})\}S(z_{t})]\\
& =E[\phi_{1}(z_{t},\beta_{0},\gamma_{0},\lambda_{0})S(z_{t})].\end{aligned}$$
Next, we show the result for $\phi_{j}(z,\beta,\gamma,\lambda)$ for $2\leq
j\leq J.$ As in the proof of Proposition 4 of Newey (1994a), for any $w_{t}$ we have$$\frac{\partial}{\partial\tau}E[w_{t}|x_{t},y_{tj}=1,\tau]=E[\frac{y_{tj}}{P_{j}(\tilde{v}_{t})}\{w_{t}-E[w_{t}|x_{t},y_{tj}=1]\}S(z_{t})|x_{t}].$$ It follows that$$\begin{aligned}
\frac{\partial E[m(z_{t},\beta_{0},\gamma_{j}(\tau),\gamma_{-j,0})]}{\partial\tau} & =-\delta E[A(x_{t})P_{vj}(\tilde{v}_{t})\frac{\partial
E[u_{1,t+1}+H_{t+1}|x_{t},y_{tj}=1,\tau]}{\partial\tau}]\\
& =-\delta\frac{\partial}{\partial\tau}E[E[A(x_{t})P_{vj}(\tilde{v}_{t})\{u_{1,t+1}+H_{t+1}\}|x_{t},y_{tj}=1,\tau]].\\
& =-\delta E[A(x_{t})P_{vj}(\tilde{v}_{t})\frac{y_{tj}}{P_{j}(\tilde{v}_{t})}\{u_{1,t+1}+H_{t+1}-\gamma_{j0}(x_{t},\beta_{0},\gamma_{1})\}S(z_{t})]\\
& =E[\phi_{j}(z_{t},\beta_{0},\gamma_{0},\lambda_{0})S(z_{t})],\end{aligned}$$ showing that the formula for $\phi_{j}$ is correct. The proof for $\phi_{J+1}$ follows similarly. *Q.E.D.*
**Proof of Theorem 3:** Given in text.
**Proof of Theorem 4:** Given in text.
**Proof of Theorem 5:** Let $\bar{\psi}(\gamma,\lambda)=E[\psi
(z_{i},\beta_{0},\gamma,\lambda)]$. Suppose that $\psi(z,\beta,\gamma
,\lambda)$ is DR. Then for any $\gamma\neq\gamma_{0},\gamma\in\Gamma$ we have$$0=\bar{\psi}(\gamma,\lambda_{0})=\bar{\psi}(\gamma_{0},\lambda_{0})=\bar{\psi
}((1-\tau)\gamma_{0}+\tau\gamma,\lambda_{0}),$$ for any $\tau.$ Therefore for any $\tau$,$$\bar{\psi}((1-\tau)\gamma_{0}+\tau\gamma,\lambda_{0})=0=(1-\tau)\bar{\psi
}(\gamma_{0},\lambda_{0})+\tau\bar{\psi}(\gamma,\lambda_{0}),$$ so that $\bar{\psi}(\gamma,\lambda_{0})$ is affine in $\gamma.$ Also by the previous equation $\bar{\psi}((1-\tau)\gamma_{0}+\tau\gamma,\lambda_{0})=0$ identically in $\tau$ so that $$\frac{\partial}{\partial\tau}\bar{\psi}((1-\tau)\gamma_{0}+\tau\gamma
,\lambda_{0})=0,$$ where the derivative with respect to $\tau$ is evaluated at $\tau=0.$ Applying the same argument switching of $\lambda$ and $\gamma$ we find that $\bar{\psi
}(\gamma_{0},\lambda)$ is affine in $\lambda$ and $\partial\bar{\psi}(\gamma_{0},(1-\tau)\lambda_{0}+\tau\lambda)/\partial\tau=0.$
Next suppose that $\bar{\psi}(\gamma,\lambda_{0})$ is affine $\gamma$ and $\partial\bar{\psi}((1-\tau)\gamma_{0}+\tau\gamma,\lambda_{0})/\partial
\tau=0.$ Then by $\bar{\psi}(\gamma_{0},\lambda_{0})=0$, for any $\gamma
\in\Gamma,$ $$\begin{aligned}
\bar{\psi}(\gamma,\lambda_{0}) & =\partial\lbrack\tau\bar{\psi}(\gamma,\lambda_{0})]/\partial\tau=\partial\lbrack(1-\tau)\bar{\psi}(\gamma_{0},\lambda_{0})+\tau\bar{\psi}(\gamma,\lambda_{0})]/\partial\tau\\
& =\partial\bar{\psi}((1-\tau)\gamma_{0}+\tau\gamma,\lambda_{0})/\partial
\tau=0.\end{aligned}$$ Switching the roles of $\gamma$ and $\lambda$ it follows analogously that $\bar{\psi}(\gamma_{0},\lambda)=0$ for all $\lambda\in\Lambda,$ so $\bar{\psi
}(\gamma,\lambda)$ is doubly robust. *Q.E.D.*
**Proof of Theorem 6:** Let $\lambda_{0}(x)=-c^{\prime}\Pi^{-1}a(x)$ so that $E[\lambda_{0}(x_{i})|w_{i}]=-c^{\prime}\Pi^{-1}\Pi p(w_{i})=-c^{\prime
}p(w_{i}).$Then integration by parts gives$$\begin{aligned}
E[m(z_{i},\beta_{0},\tilde{\gamma})] & =E[c^{\prime}p(w_{i})\{\tilde{\gamma
}(w_{i})-\gamma_{0}(w_{i})\}]=-E[\gamma_{0}(x_{i})\{\tilde{\gamma}(w_{i})-\gamma_{0}(w_{i})\}]\\
& =E[\gamma_{0}(x_{i})\{y_{i}-\tilde{\gamma}(w_{i})\}]=-c^{\prime}\Pi
^{-1}E[a(x_{i})\{y_{i}-\tilde{\gamma}(w_{i})\}]=0.\text{ }Q.E.D.\end{aligned}$$
**Proof of Theorem 7:** If $\lambda_{0}$ is identified then $m(z,\beta,\bar{\gamma},\lambda_{0})$ is identified for every $\beta$. By DR$$E[m(z_{i},\beta,\bar{\gamma},\lambda_{0})]=0$$ at $\beta=\beta_{0}$ and by assumption this is the only $\beta$ where this equation is satisfied. *Q.E.D.*
**Proof of Corollary 8:** Given in text.
**Proof of Theorem 9:** Note that for $\rho_{i}=\rho(z_{i},\beta
_{0},\gamma_{0}),$$$\bar{\psi}(\gamma_{0},(1-\tau)\lambda_{0}+\tau\lambda)]=(1-\tau)E[\lambda
_{0}(x_{i})\rho_{i}]+\tau E[\lambda(x_{i})\rho_{i}]=0. \label{th9proof}$$ Differentiating gives the second equality in eq. (\[lrdef2\]). Also, for $\Delta=\gamma-\gamma_{0},$$$\frac{\partial\bar{\psi}((1-\tau)\gamma_{0}+\tau\gamma,\lambda_{0})}{\partial\tau}=E[\lambda_{0}(x_{i})\bar{\rho}(x_{i},\Delta)]=0,$$ giving the first equality in eq. (\[lrdef2\]). *Q.E.D.*
**Proof of Theorem 10:** The first equality in eq. (\[th9proof\]) of the proof of Theorem 9 shows that $\bar{\psi}(\gamma_{0},\lambda)$ is affine in $\lambda$. Also,$$\bar{\psi}((1-\tau)\gamma_{0}+\tau\gamma,\lambda_{0})=E[\lambda_{0}(x_{i})\{(1-\tau)\rho(z_{i},\beta_{0},\gamma_{0})+\tau\rho(z_{i},\beta
_{0},\gamma)\}]=(1-\tau)\bar{\psi}(\gamma_{0},\lambda_{0})+\tau\bar{\psi
}(\gamma,\lambda_{0}),$$ so that $\bar{\psi}(\gamma,\lambda_{0})$ is affine in $\gamma.$ The conclusion then follows by Theorem 5. *Q.E.D.*
**Proof of Theorem 11:** To see that $\tilde{\lambda}^{\Sigma^{\ast}}(x_{i},\lambda^{\ast})\Sigma^{\ast}(x_{i})^{-1}$ minimizes the asymptotic variance note that for any orthogonal instrumental variable matrix $\lambda_{0}(x),$ by the rows of $\lambda_{\beta}(x_{i})-\tilde{\lambda
}^{\Sigma^{\ast}}(x_{i},\lambda_{\beta})$ being in $\bar{\Lambda}_{\gamma},$ $$M=E[\lambda_{0}(x_{i})\lambda_{\beta}(x_{i})^{\prime}]=E[\lambda_{0}(x_{i})\tilde{\lambda}^{\Sigma^{\ast}}(x_{i},\lambda_{\beta})^{\prime
}]=E[\lambda_{0}(x_{i})\rho_{i}\rho_{i}^{\prime}\Sigma^{\ast}(x_{i})^{-1}\tilde{\lambda}^{\Sigma^{\ast}}(x_{i},\lambda_{\beta})^{\prime}].$$ Since the instruments are orthogonal the asymptotic variance matrix of the GMM estimator with $\hat{W}\overset{p}{\longrightarrow}W$ is the same as if $\hat{\gamma}=\gamma_{0}.$ Define $m_{i}=M^{\prime}W\lambda_{0}(x_{i})\rho
_{i}$ and $m_{i}^{\ast}=\tilde{\lambda}^{\Sigma^{\ast}}(x_{i},\lambda_{\beta
})\Sigma^{\ast}(x_{i})^{-1}\rho_{i}.$ The asymptotic variance of the GMM estimator for orthogonal instruments $\lambda_{0}(x)$ is$$(M^{\prime}WM)^{-1}M^{\prime}WE[\lambda_{0}(x_{i})\rho_{i}\rho_{i}^{\prime
}\lambda_{0}(x_{i})^{\prime}]WM(M^{\prime}WM)^{-1}=(E[m_{i}m_{i}^{\ast\prime
}])^{-1}E[m_{i}m_{i}^{\prime}](E[m_{i}m_{i}^{\ast}])^{-1\prime}.$$ The fact that this matrix is minimized in the positive semidefinite sense for $m_{i}=m_{i}^{\ast}$ is well known, e.g. see Newey and McFadden (1994). *Q.E.D.*
The following result is useful for the results of Section 7:
<span style="font-variant:small-caps;">Lemma A1:</span> *If Assumption 4 is satisfied then* $\hat{R}_{1}\overset{p}{\longrightarrow}0.$ *If Assumption 5 is satisfied then* $\hat{R}_{2}\overset{p}{\longrightarrow}0.$
Proof: Define $\hat{\Delta}_{i\ell}=m(z_{i},\hat{\gamma}_{\ell})-m(z_{i},\gamma_{0})-\bar{m}(\hat{\gamma}_{\ell})$ for $i\in I_{\ell}$ and let $Z_{\ell}^{c}$ denote the observations $z_{i}$ for $i\notin I_{\ell}$. Note that $\hat{\gamma}_{\ell}$ depends only on $Z_{\ell}^{c}$. By construction and independence of $Z_{\ell}^{c}$ and $z_{i},i\in I_{\ell}$ we have $E[\hat{\Delta}_{i\ell}|Z_{\ell}^{c}]=0.$ Also by independence of the observations, $E[\hat{\Delta}_{i\ell}\hat{\Delta}_{j\ell}|Z_{\ell}^{c}]=0$ for $i,j\in I_{\ell}.$ Furthermore, for $i\in I_{\ell}$ $E[\hat{\Delta}_{i\ell
}^{2}|Z_{\ell}^{c}]\leq\int[m(z,\hat{\gamma}_{\ell})-m(z,\gamma_{0})]^{2}F_{0}(dz)$. Then we have $$\begin{aligned}
E[\left( \frac{1}{\sqrt{n}}\sum_{i\in I_{\ell}}\hat{\Delta}_{i\ell}\right)
^{2}|Z_{\ell}^{c}] & =\frac{1}{n}E[\left( \sum_{i\in I_{\ell}}\hat{\Delta
}_{i\ell}\right) ^{2}|Z_{\ell}^{c}]=\frac{1}{n}\sum_{i\in I_{\ell}}E[\hat{\Delta}_{i\ell}^{2}|Z_{\ell}^{c}]\\
& \leq\int[m(z,\hat{\gamma}_{\ell})-m(z,\gamma_{0})]^{2}F_{0}(dz)\overset{p}{\longrightarrow}0.\end{aligned}$$ The conditional Markov inequality then implies that $\sum_{i\in I_{\ell}}\hat{\Delta}_{i\ell}/\sqrt{n}\overset{p}{\longrightarrow}0.$ The analogous results also hold for $\hat{\Delta}_{i\ell}=\phi(z_{i},\hat{\gamma}_{\ell
},\lambda_{0})-\phi(z_{i},\gamma_{0},\lambda_{0})-\bar{\phi}(\hat{\gamma
}_{\ell},\lambda_{0})$ and $\hat{\Delta}_{i\ell}=\phi(z_{i},\gamma_{0},\hat{\lambda}_{\ell})-\phi(z_{i},\gamma_{0},\lambda_{0})-\bar{\phi}(\gamma_{0},\hat{\lambda}_{\ell})$. Summing across these three terms and across $\ell=1,...,L$ gives the first conclusion.
For the second conclusion, note that under the first hypothesis of Assumption 5,$$\begin{aligned}
& E[\left\vert \frac{1}{\sqrt{n}}\sum_{i\in I_{\ell}}[\phi_{j}(z_{i},\hat{\gamma}_{\ell},\hat{\lambda}_{\ell})-\phi_{j}(z_{i},\gamma_{0},\hat{\lambda}_{\ell})-\phi_{j}(z_{i},\hat{\gamma}_{\ell},\lambda_{0})+\phi_{j}(z_{i},\gamma_{0},\lambda_{0})]\right\vert |Z_{\ell}^{c}]\\
& \leq\frac{1}{\sqrt{n}}\sum_{i\in I_{\ell}}E[\left\vert \phi_{j}(z_{i},\hat{\gamma}_{\ell},\hat{\lambda}_{\ell})-\phi_{j}(z_{i},\gamma_{0},\hat{\lambda}_{\ell})-\phi_{j}(z_{i},\hat{\gamma}_{\ell},\lambda_{0})+\phi_{j}(z_{i},\gamma_{0},\lambda_{0})\right\vert |Z_{\ell}^{c}]\\
& \leq\sqrt{n}\int\left\vert \phi_{j}(z,\hat{\gamma}_{\ell},\hat{\lambda
}_{\ell})-\phi_{j}(z,\gamma_{0},\hat{\lambda}_{\ell})-\phi_{j}(z,\hat{\gamma
}_{\ell},\lambda_{0})+\phi_{j}(z_{i},\gamma_{0},\lambda_{0})\right\vert
F_{0}(dz)\overset{p}{\longrightarrow}0,\end{aligned}$$ so $\hat{R}_{2}\overset{p}{\longrightarrow}0$ follows by the conditional Markov and triangle inequalities. The second hypothesis of Assumption 5 is just $\hat{R}_{2}\overset{p}{\longrightarrow}0.$ $Q.E.D.$
**Proof of Lemma 12**: By Assumption 1 and the hypotheses that $\hat{\gamma}_{i}\in\Gamma$ and $\hat{\lambda}_{i}\in\Lambda$ we have $\hat
{R}_{3}=\hat{R}_{4}=0.$ By Lemma A1 we have $\hat{R}_{1}\overset{p}{\longrightarrow}0$ and $\hat{R}_{2}\overset{p}{\longrightarrow}0.$ The conclusion then follows by the triangle inequality. $Q.E.D.$
**Proof of Theorem 13:** Note that for $\varepsilon=y-\gamma_{0}(w)$ $$\begin{aligned}
\phi(z,\hat{\gamma},\lambda_{0})-\phi(z,\gamma_{0},\lambda_{0}) &
=\lambda_{0}(x)[\hat{\gamma}(w)-\gamma_{0}(w)],\\
\phi(z,\gamma_{0},\hat{\lambda})-\phi(z,\gamma_{0},\lambda_{0}) &
=[\hat{\lambda}(x)-\lambda_{0}(x)]\varepsilon,\\
\phi(z,\hat{\gamma}_{\ell},\hat{\lambda}_{\ell})-\phi(z,\gamma_{0},\hat{\lambda}_{\ell})-\phi(z,\hat{\gamma}_{\ell},\lambda_{0})+\phi
_{j}(z,\gamma_{0},\lambda_{0}) & =-[\hat{\lambda}(x)-\lambda_{0}(x)][\hat{\gamma}(x)-\gamma_{0}(x)].\end{aligned}$$ The first part of Assumption 4 ii) then follows by$$\begin{aligned}
\int[\phi(z,\hat{\gamma}_{\ell},\lambda_{0})-\phi(z,\gamma_{0},\lambda
_{0})]^{2}F_{0}(dz) & =\int\lambda_{0}(x)^{2}[\hat{\gamma}(w)-\gamma
_{0}(w)]^{2}F_{0}(dz)\\
& \leq C\int[\hat{\gamma}(w)-\gamma_{0}(w)]^{2}F_{0}(dz)\overset{p}{\longrightarrow}0.\end{aligned}$$ The second part of Assumption 4 ii) follows by$$\begin{aligned}
\int[\phi(z,\gamma_{0},\hat{\lambda}_{\ell})-\phi(z,\gamma_{0},\lambda
_{0})]^{2}F_{0}(dz) & =\int[\hat{\lambda}_{\ell}(x)-\lambda_{0}(x)]^{2}\varepsilon^{2}F_{0}(dz)\\
& =\int\left[ \hat{\lambda}_{\ell}(x)-\lambda_{0}(x)\right] ^{2}E[\varepsilon^{2}|x]F_{0}(dz)\\
& \leq C\int\left[ \hat{\lambda}_{\ell}(x)-\lambda_{0}(x)\right] ^{2}F_{0}(dz)\overset{p}{\longrightarrow}0.\end{aligned}$$ Next, note that by the Cauchy-Schwartz inequality, $$\begin{aligned}
& \sqrt{n}\int|\phi(z,\hat{\gamma}_{\ell},\hat{\lambda}_{\ell})-\phi
(z,\gamma_{0},\hat{\lambda}_{\ell})-\phi(z,\hat{\gamma}_{\ell},\lambda
_{0})+\phi(z,\gamma_{0},\lambda_{0})|F_{0}(dz)\\
& =\sqrt{n}\int\left\vert [\hat{\lambda}_{\ell}(x)-\lambda_{0}(x)][\hat
{\gamma}_{\ell}(w)-\gamma_{0}(w)]\right\vert F_{0}(dx)\\
& \leq\sqrt{n}\{\int[\hat{\lambda}_{\ell}(x)-\lambda_{0}(x)]^{2}F_{0}(dx)\}^{1/2}\{\int[\hat{\gamma}_{\ell}(w)-\gamma_{0}(w)]^{2}F_{0}(dw)\}^{1/2}.\end{aligned}$$ Then the first rate condition of Assumption 5 holds under the first rate condition of Theorem 13 while the second condition of Assumption 5 holds under the last hypothesis of Theorem 13. Then eq. (\[no effec\]) holds by Lemma 12, and the conclusion by rearranging the terms in eq. (\[no effec\]). *Q.E.D.*
**Proof of Lemma 14:** Follows by Lemma A1 and the triangle inequality. *Q.E.D.*
**Proof of Lemma 15:** Let $\hat{M}(\beta)=\partial\hat{\psi}(\beta)/\partial\beta$ when it exists, $\tilde{M}_{\ell}=n^{-1}\sum_{i\in
I_{\ell}}\partial\psi(z_{i},\beta_{0},\hat{\gamma}_{\ell},\hat{\lambda}_{\ell
})/\partial\beta,$ and $\bar{M}_{\ell}=n^{-1}\sum_{i\in I_{\ell}}\partial
\psi(z_{i},\beta_{0},\gamma_{0},\lambda_{0})/\partial\beta.$ By the law of large numbers, and Assumption 5 iii), $\sum_{\ell=1}^{L}\bar{M}_{\ell
}\overset{p}{\longrightarrow}M.$ Also, by condition iii) for each $j$ and $k,$ $$E[|\tilde{M}_{\ell jk}-\bar{M}_{\ell jk}||Z^{\ell}]\leq\int\left\vert
\partial\psi_{j}(z,\beta_{0},\hat{\gamma}_{\ell},\hat{\lambda}_{\ell
})/\partial\beta_{k}-\partial\psi_{j}(z,\beta_{0},\gamma_{0},\lambda
_{0})/\partial\beta_{k}\right\vert F_{0}(dz)\overset{p}{\longrightarrow}0.$$ Then by the conditional Markov inequality, for each $\ell,$ $$\tilde{M}_{\ell}-\bar{M}_{\ell}\overset{p}{\longrightarrow}0.$$ It follows by the triangle inequality that $\sum_{\ell=1}^{L}\tilde{M}_{\ell
}\overset{p}{\longrightarrow}M.$ Also, with probability approaching one we have for any $\bar{\beta}\overset{p}{\longrightarrow}\beta_{0}$$$\left\Vert \hat{M}(\bar{\beta})-\sum_{\ell=1}^{L}\tilde{M}_{\ell}\right\Vert
\leq\left( \frac{1}{n}\sum_{i=1}^{n}d(z_{i})\right) \left\Vert \bar{\beta
}-\beta_{0}\right\Vert ^{\zeta^{\prime}}=O_{p}(1)o_{p}(1)\overset{p}{\longrightarrow}0.$$ The conclusion then follows by the triangle inequality. *Q.E.D.*
**Proof of Theorem 16:** The conclusion follows in a standard manner from the conclusions of Lemmas 14 and 15. *Q.E.D.*
**Proof of Theorem 17:** Let $\hat{\psi}_{i}=\psi(z_{i},\hat{\beta},\hat{\gamma}_{\ell},\hat{\lambda}_{\ell})$ and $\psi_{i}=\psi(z_{i},\beta
_{0},\gamma_{0},\lambda_{0}).$ By standard arguments (e.g. Newey, 1994), it suffices to show that $\sum_{i=1}^{n}\left\Vert \hat{\psi}_{i}-\psi
_{i}\right\Vert ^{2}/n\overset{p}{\longrightarrow}0.$ Note that$$\begin{aligned}
\hat{\psi}_{i}-\psi_{i} & =\sum_{j=1}^{5}\hat{\Delta}_{ji},\hat{\Delta}_{1i}=\psi(z_{i},\hat{\beta},\hat{\gamma}_{\ell},\hat{\lambda}_{\ell})-\psi(z_{i},\beta_{0},\hat{\gamma}_{\ell},\hat{\lambda}_{\ell}),\hat{\Delta
}_{2i}=m(z_{i},\beta_{0},\hat{\gamma}_{\ell})-m(z_{i},\beta_{0},\gamma_{0}),\\
\hat{\Delta}_{3i} & =\phi(z_{i},\hat{\gamma}_{\ell},\lambda_{0})-\phi
(z_{i},\gamma_{0},\lambda_{0}),\hat{\Delta}_{4i}=\phi(z_{i},\gamma_{0},\hat{\lambda}_{\ell})-\phi(z_{i},\gamma_{0},\lambda_{0}),\\
\hat{\Delta}_{5i} & =\phi(z_{i},\hat{\gamma}_{\ell},\hat{\lambda}_{\ell
})-\phi(z_{i},\hat{\gamma}_{\ell},\lambda_{0})-\phi(z_{i},\gamma_{0},\hat{\lambda}_{\ell})+\phi(z_{i},\gamma_{0},\lambda_{0}).\end{aligned}$$ By standard arguments it suffices to show that for each $j$ and $\ell,$ $$\frac{1}{n}\sum_{i\in I_{\ell}}\left\Vert \hat{\Delta}_{ji}\right\Vert
^{2}\overset{p}{\longrightarrow}0. \label{var conv}$$ For $j=1$ it follows by a mean value expansion and Assumption 7 with $E[b(z_{i})^{2}]<\infty$ that$$\frac{1}{n}\sum_{i\in I_{\ell}}\left\Vert \hat{\Delta}_{1i}\right\Vert
^{2}=\frac{1}{n}\sum_{i\in I_{\ell}}\left\Vert \frac{\partial}{\partial\beta
}\psi(z_{i},\bar{\beta},\hat{\gamma}_{\ell},\hat{\lambda}_{\ell})(\hat{\beta
}-\beta)\right\Vert ^{2}\leq\frac{1}{n}\left( \sum_{i\in I_{\ell}}b(z_{i})^{2}\right) \left\Vert \hat{\beta}-\beta\right\Vert ^{2}\overset{p}{\longrightarrow}0,$$ where $\bar{\beta}\,$is a mean value that actually differs from row to row of $\partial\psi(z_{i},\bar{\beta},\hat{\gamma}_{\ell},\hat{\lambda}_{\ell
})/\partial\beta$. For $j=2$ note that by Assumption 4,$$E[\frac{1}{n}\sum_{i\in I_{\ell}}\left\Vert \hat{\Delta}_{2i}\right\Vert
^{2}|Z^{\ell}]\leq\int\left\Vert m(z,\beta_{0},\hat{\gamma}_{\ell})-m(z,\beta_{0},\gamma_{0})\right\Vert ^{2}F_{0}(dz)\overset{p}{\longrightarrow}0,$$ so eq. (\[var conv\]) holds by the conditional Markov inequality. For $j=3$ and $j=4$ eq. (\[var conv\]) follows similarly. For $j=5$, it follows from the hypotheses of Theorem 17 that$$E[\frac{1}{n}\sum_{i\in I_{\ell}}\left\Vert \hat{\Delta}_{5i}\right\Vert
^{2}|Z^{\ell}]\leq\int\left\Vert \phi(z,\hat{\gamma}_{\ell},\hat{\lambda
}_{\ell})-\phi(z,\gamma_{0},\hat{\lambda}_{\ell})-\phi(z,\hat{\gamma}_{\ell
},\lambda_{0})+\phi(z,\gamma_{0},\lambda_{0})\right\Vert ^{2}F_{0}(dz)\overset{p}{\longrightarrow}0.$$ Then eq. (\[var conv\]) holds for $j=5$ by the conditional Markov inequality. *Q.E.D.*
Appendix B: Local Robustness and Derivatives of Expected Moments.
=================================================================
In this Appendix we give conditions sufficient for the LR property of equation (\[lrdef\]) to imply the properties in equations (\[lrdef2\]) and (\[nlremainder\]). As discussed following equation (\[nlremainder\]), it may be convenient when specifying regularity conditions for specific moment functions to work directly with (\[lrdef2\]) and/or (\[nlremainder\]).
<span style="font-variant:small-caps;">Assumption B1:</span> *There are linear sets* $\Gamma$ *and* $\Lambda$ *and a set* $G$ *such that i)* $\bar{\psi}(\gamma,\lambda)$ *is Frechet differentiable at* $(\gamma_{0},\lambda_{0});$ *ii) for all* $G\in$ ** $G$ *the vector* $(\gamma(F_{\tau}),\lambda(F_{\tau}))$ *is Frechet differentiable at* $\tau=0;$ *iii) the closure of* $\{\partial(\gamma(F_{\tau}),\lambda(F_{\tau}))/\partial\tau:G\in$ ** $G\}$ *is* $\Gamma\times\Lambda$*.*
<span style="font-variant:small-caps;">Theorem B1:</span> *If Assumption B1 is satisfied and equation (\[lrdef\]) is satisfied for all* $G\in$ ** $\mathcal{G}$ *then equation (\[lrdef2\]) is satisfied.*
Proof: Let $\bar{\psi}^{\prime}(\gamma,\lambda)$ denote the Frechet derivative of $\bar{\psi}(\gamma,\lambda)$ at $(\gamma_{0},\lambda_{0})$ in the direction $(\gamma,\lambda),$ which exists by i). By ii), the chain rule for Frechet derivatives (e.g. Proposition 7.3.1 of Luenberger, 1969), and by eq. *(\[lrdef\])* it follows that for $(\Delta_{\gamma}^{G},\Delta_{\lambda}^{G})=\partial(\gamma(F_{\tau}),\lambda(F_{\tau}))/\partial\tau,$$$\bar{\psi}^{\prime}(\Delta_{\gamma}^{G},\Delta_{\lambda}^{G})=\frac
{\partial\bar{\psi}(\gamma(F_{\tau}),\lambda(F_{\tau}))}{\partial\tau}=0.$$ By $\bar{\psi}^{\prime}(\gamma,\lambda)$ being a continuous linear function and iii) it follows that $\bar{\psi}^{\prime}(\gamma,\lambda)=0$ for all $(\gamma,\lambda)\in\Gamma\times\Lambda.$ Therefore, for any $\gamma\in\Gamma$ and $\lambda\in\Lambda,$$$\bar{\psi}^{\prime}(\gamma-\gamma_{0},0)=0,\bar{\psi}^{\prime}(0,\lambda
-\lambda_{0})=0.$$ Equation *(\[lrdef2\])* then follows by i). *Q.E.D.*
<span style="font-variant:small-caps;">Theorem B2:</span> *If equation (\[lrdef2\]) is satisfied and in addition* $\bar{\psi}(\gamma,\lambda_{0})$ *and* $\bar{\psi}(\gamma
_{0},\lambda)$ *are twice Frechet differentiable in open sets containing* $\gamma_{0}$ *and* $\lambda_{0}$ *respectively with bounded second derivative then equation* (\[nlremainder\]) *is satisfied.*
Proof: Follows by Proposition 7.3.3 of Luenberger (1969). *Q.E.D.*
Appendix C: Doubly Robust Moment Functions for Orthogonality Conditions
=======================================================================
In this Appendix we generalize the DR estimators for conditional moment restrictions to orthogonality conditions for a general residual $\rho
(z,\gamma)$ that is affine in $\gamma$ but need not have the form $y-\gamma(w).$
<span style="font-variant:small-caps;">Assumption C1:</span> *There are linear sets* $\Gamma$ and $\Lambda$ *of functions* $\lambda(x)$ *and* $\gamma(w)$ *that are closed in mean square such that i) For any* $\gamma,\tilde{\gamma}\in\Gamma$ and scalar $\tau,$ $E[\rho(z_{i},\gamma)^{2}]<\infty$ and $\rho(z,(1-\tau
)\gamma+\tau\tilde{\gamma})=(1-\tau)\rho(z,\gamma)+\tau\rho(z,\tilde{\gamma})$ ; *ii)* $E[\lambda(x_{i})\rho(z_{i},\gamma_{0})]=0$ for all $\lambda
\in\Lambda;$ *iii) there exists* $\lambda_{0}\in\Lambda$ *such that* $E[m(z_{i},\beta_{0},\gamma)]=-E[\lambda_{0}(x_{i})\rho(z_{i},\gamma
)]$ *for all* $\gamma\in\Gamma.$
Assumption C1 ii) could be thought of as an identification condition for $\gamma_{0}$. For example, if $\Lambda$ is all functions of $x_{i}$ with finite mean square then ii) is $E[\rho(z_{i},\gamma_{0})|x_{i}]=0,$ the nonparametric conditional moment restriction of Newey and Powell (2003) and Newey (1991). Assumption C1 iii) also has an interesting interpretation. Let $\Pi(a)(x_{i})$ denote the orthogonal mean-square projection of a random variable $a(z_{i})$ with finite second moment on $\Gamma.$ Then by ii) and iii) we have$$\begin{aligned}
E[m(z_{i},\beta_{0},\gamma)] & =-E[\lambda_{0}(x_{i})\rho(z_{i},\gamma)]=E[\lambda_{0}(x_{i})\Pi(\rho(\gamma))(x_{i})]\\
& =E[\lambda_{0}(x_{i})\{\Pi(\rho(\gamma))(x_{i})-\Pi(\rho(\gamma_{0}))(x_{i})\}]\\
& =E[\lambda_{0}(x_{i})\{\Pi(\rho(\gamma)-\rho(\gamma_{0}))(x_{i})\}].\end{aligned}$$ Here we see that $E[m(z_{i},\beta_{0},\gamma)]$ is a linear, mean-square continuous function of $\Pi(\rho(\gamma)-\rho(\gamma_{0}))(x_{i}).$ The Riesz representation theorem will also imply that if $E[m(z_{i},\beta_{0},\gamma)]$ is a linear, mean-square continuous function of $\Pi(\rho(\gamma)-\rho
(\gamma_{0}))(x_{i})$ then $\lambda_{0}(x)$ exists satisfying Assumption C1 ii). For the case where $w_{i}=x_{i}$ this mean-square continuity condition is necessary for existence of a root-n consistent estimator, as in Newey (1994) and Newey and McFadden (1994). We conjecture that when $w_{i}$ need not equal $x_{i}$ this condition generalizes Severini and Tripathi’s (2012) necessary condition for existence of a root-n consistent estimator of $\beta_{0}$.
Noting that Assumptions 1 ii) and iii) are the conditions for double robustness we have
<span style="font-variant:small-caps;">Theorem C1:</span> *If Assumption C1 is satisfied then* $\psi
(z,\beta,\gamma,\lambda)=m(z,\beta,\gamma)+\lambda(x)\rho(z,\gamma)$ *is doubly robust.*
It is interesting to note that $\lambda_{0}(x)$ satisfying Assumption C1 iii) need not be unique. When the closure of $\{\Pi(\rho(\gamma))(x_{i}):\gamma
\in\Gamma\}$ is not all of $\Lambda$ then there will exist $\tilde{\lambda}\in\Lambda$ such that $\tilde{\lambda}\neq0$ and $$E[\tilde{\lambda}(x_{i})\rho(z_{i},\gamma)]=E[\tilde{\lambda}(x_{i})\Pi
(\rho(\gamma))(x_{i})]=0\text{ for all }\gamma\in\Gamma.$$ In that case Assumption C1 iii) will also be satisfied for $\lambda_{0}(x_{i})+\tilde{\lambda}(x_{i}).$ We can think of this case as one where $\gamma_{0}$ is overidentified, similarly to Chen and Santos (2015). As discussed in Ichimura and Newey (2017), the different $\lambda_{0}(x_{i})$ would correspond to different first step estimators.
The partial robustness results of the last Section can be extended to the orthogonality condition setting of Assumption C1. Let $\Lambda^{\ast}$ be a closed linear subset of $\Lambda,$ such as finite dimensional linear set and let $\gamma^{\ast}$ be such that $E[\lambda(x_{i})\rho(z_{i},\gamma^{\ast
})]=0$ for all $\lambda\in\Lambda^{\ast}$. Note that if $\lambda_{0}\in
\Lambda^{\ast}$ it follows by Theorem C1 that$$E[m(z_{i},\beta_{0},\gamma^{\ast})]=-E[\lambda_{0}(x_{i})\rho(z_{i},\gamma^{\ast})]=0.$$
<span style="font-variant:small-caps;">Theorem C2:</span> *If* $\Lambda^{\ast}$ *is a closed linear subset of* $\Lambda$*,* $E[\lambda(x_{i})\rho(z_{i},\gamma^{\ast})]=0$ *for all* $\lambda\in\Lambda^{\ast}$*, and Assumption C2 iii) is satisfied with* $\lambda_{0}\in\Lambda^{\ast}$ *then*$$E[m(z_{i},\beta_{0},\gamma^{\ast})]=0.$$
$.$
Appendix D: Regularity Conditions for Plug-in Estimators
========================================================
In this Appendix we formulate regularity conditions for root-n consistency and asymptotic normality of the plug-in estimator $\tilde{\beta}$ as described in Section 2, where $m(z,\beta,\gamma)$ need not be LR. These conditions are based on Assumptions 4-6 applied to the influence adjustment $\phi
(z,\gamma,\lambda)$ corresponding to $m(z,\beta,\gamma)$ and $\hat{\gamma}.$ For this purpose we treat $\hat{\lambda}$ as any object that can approximate $\lambda_{0}(x),$ not just as an estimator of $\lambda_{0}.$
<span style="font-variant:small-caps;">Theorem D1:</span> *If Assumptions 4-6 are satisfied, Assumption 7* is satisfied with $m(z,\beta,\gamma)$ replacing $\psi(z,\beta,\gamma
,\lambda),$ ** $\tilde{\beta}\overset{p}{\longrightarrow}\beta_{0},$ ** $\hat{W}\overset{p}{\longrightarrow}W$*,* $M^{\prime}WM$ *is nonsingular,* $E[\left\Vert \psi(z_{i},\beta_{0},\gamma
_{0},\lambda_{0})\right\Vert ^{2}]<\infty,$ *and*$$\hat{R}_{5}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\phi(z_{i},\hat{\gamma}_{i},\hat{\lambda}_{i})\overset{p}{\longrightarrow}0,$$ *then for* $\Omega=E[\psi(z_{i},\beta_{0},\gamma_{0},\lambda_{0})\psi(z_{i},\beta_{0},\gamma_{0},\lambda_{0})^{\prime}],$$$\sqrt{n}(\hat{\beta}-\beta_{0})\overset{d}{\longrightarrow}N(0,V),V=(M^{\prime
}WM)^{-1}M^{\prime}W\Omega WM(M^{\prime}WM)^{-1}.$$
The condition $\hat{R}_{5}\overset{p}{\longrightarrow}0$ was discussed in Section 7. It is interesting to note that $\hat{R}_{5}\overset{p}{\longrightarrow}0$ appears to be a complicated condition that seems to depend on details of the estimator $\hat{\gamma}_{i}$ in a way that Assumptions 4-7 do not. In this way the regularity conditions for the LR estimator seem to be more simple and general than those for the plug-in estimator.
Acknowledgements
Whitney Newey gratefully acknowledges support by the NSF. Helpful comments were provided by M. Cattaneo, B. Deaner, J. Hahn, M. Jansson, Z. Liao, A. Pakes, R. Moon, A. de Paula, V. Semenova, and participants in seminars at Cambridge, Columbia, Cornell, Harvard-MIT, UCL, USC, Yale, and Xiamen. B. Deaner provided capable research assistance.
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|
Four-ever? Competition remedies in the audit market
Oxera
In light of recent accounting scandals, there are widespread calls for the UK competition authority to re-examine the audit market. Yet spending a substantial amount of resources on a market investigation, and concluding once again that there is a competition problem, is of little value if a suitable remedy cannot be found. A break-up of the Big Four is perceived by many as a necessary and long-awaited intervention, but is it the right solution? And if not, what would be an alternative remedy?
The UK audit market has gone through some turmoil recently.[1] This month the Financial Reporting Council (FRC), which regulates UK audit, announced a deterioration in audit quality across the ‘Big Four’ firms (KPMG, PwC, Deloitte and EY) compared with the previous year. Most notably, the FRC noted that 50% of KPMG’s FTSE 350 audits failed to reach the FRC’s standard for audit quality.[2] At a global level, the International Forum of Independent Audit Regulators found significant problems in 40% of the 918 audits of listed public interest entities that it inspected last year.[3]
The recent audit failures uncovered by regulators are hardly trivial. In Miller Energy the US Securities and Exchange Commission found that KPMG had overvalued certain assets by more than 100 times.[4] In BHS the FRC noted that PwC had signed off the accounts just days before the company was sold for £1.[5] In the more recent case of Carillion, equity analysts appeared unaware of the warning signs that might have been flagged by a good audit.[6]
These market outcomes in audit services are unsatisfactory from a policy perspective. The Big Four’s joint market share in FTSE 350 audit has been close to 100% for many years, and the Big Four likewise dominate the audit of large companies across the world. It is this high market concentration that is frequently blamed for the poor outcomes,[7] and regulators and competition authorities across the world have raised concerns about concentration ever since the collapse of Arthur Andersen in 2002. This year, two UK Parliamentary Committees have called for a new competition investigation by the Competition and Markets Authority (CMA) that ‘should explicitly include consideration of both breaking up the Big Four into more audit firms, and detaching audit arms from those providing other professional services’.[8] The Chief Executive Officer of the FRC and the CEO of PwC have both expressed support for the idea of having the CMA study the audit market afresh.[9]
Previous remedies in the audit market
The audit market is effectively dominated at the top end by the Big Four, and despite turmoil in financial markets the audit market structure has remained largely unchanged since 2002.[10] Concerns emanating from the high concentration include a lack of choice, a lack of innovation, higher audit fees, conflicts of interest, a lack of independence that weakens auditor professional scepticism, a systemic risk if one Big Four firm should fail, and, above all, poor-quality audit reducing the credibility and reliability of audited financial statements for the world’s largest companies.[11]
The previous investigation by the UK Competition Commission (CC), predecessor to the CMA, put forward a package of seven remedies, the most significant of which was a requirement that FTSE 350 companies put their audit out to tender at least every ten years (‘mandatory tendering’). Shortly thereafter, the EU introduced rules that obliged listed companies to switch their auditor (‘mandatory rotation’) every 20 years.[12] At the conclusion of the previous market investigation the CC expressed confidence in its package of remedies, noting that they should ‘increase choice’ and provide a ‘substantially improved environment for competition’.[13] The CC’s remedies package did not include any structural remedies.
The CC and EU remedies have not solved the problem of attracting more competition from outside the Big Four.[14] Indeed, the leading non-Big Four firms, Grant Thornton and BDO, between them have fewer FTSE 350 clients than before the regulatory interventions. In 2013, just before the new measures to boost competition were enacted, Grant Thornton had six FTSE 350 audit clients. In 2016, this number was unchanged. But in 2018 the firm said that it would exit the market for large audits.[15] In 2013 BDO had eight FTSE 350 clients, falling to five in 2016.[16] The previous rule changes are therefore widely perceived to have failed to remedy concerns over market concentration. The Big Four accountancy firms still audit 97% of FTSE 350 companies, a similar rate to that found by Oxera[17] in its 2006 market study for the FRC.[18]
What could structural remedies achieve?
Vertical separation
There are different types of structural remedies. Vertical separation of the Big Four firms into audit and non-audit services would not increase the basic number of firms participating in the FTSE 350 audit market, but it would increase the effective choice for many companies that have non-audit relationships with Big Four audit firms. These relationships can preclude, whether legally or in terms of company perception,[19] considering all four current audit firms as viable substitute auditors.[20]
Vertical separation would also be oriented towards audit quality, removing the conflicts of interest that can arise when the auditor also supplies valuable non-audit services. Yet the idea was not popular among investors at the time of the previous competition investigation. In 2012, an Oxera investor survey report found that ‘almost all investors surveyed do not want to see structural separation of the Big Four firms into audit and non-audit activities.’[21]
Horizontal separation
Horizontal separation of the Big Four firms would immediately improve choice in the sense of seeing more than four firms in the market, and also choice in terms of seeing several non-conflicted audit firms in every audit tender. Such a separation would therefore also, in general terms, improve competition. It could also serve audit quality by reducing the number of instances where a company involved in a complex transaction cannot realistically find an adviser that is not subject to some conflict of interest.
In the case of Carillion, PwC acted as the company’s pensions consultant (2002–17), then switched to advising the pension scheme trustees on Carillion’s restructuring proposals (from July 2017), and was finally appointed by the government to help manage the defunct Carillion after its collapse (from January 2018).[22] It would appear that PwC was the only viable choice to advise on Carillion’s insolvency, because it was the only Big Four firm that did not have active contracts with Carillion at the time of Carillion’s demise.[23] Expanding the market from a ‘Big Four’ to a ‘Large 6’ seems attractive in the face of such apparent conflicts, but realistically it would be a very difficult exercise if the aim is to create a ‘Large 6’ group of firms of similar size with similar international networks.
Would a break-up increase audit quality?
Audits are for the protection of investors against false accounting by a company’s management. The starting point is therefore that the true customer of audit, the investor, is not the procurer of audit services. This alone creates an environment in which market failures may be expected.
But why does audit quality fall short? Boeing and Airbus, Coca-Cola and Pepsi, and the Silicon Valley giants all operate in concentrated markets—but it seems highly unlikely that half of new aeroplanes, or soft drinks cans, possess substantial errors. Market concentration per se does not entail a poor-quality product: even a monopolist will have regard to product quality, knowing that if its product is faulty the financial consequences of fines and compensating consumers will typically be severe.
In equilibrium, a firm would only produce faulty items to the extent that it is rational to do so—i.e. if errors cannot be detected or if the financial consequences of errors are insubstantial. It seems to be widely accepted that audit quality is below the level demanded by investors, on whose behalf the audit is undertaken. The economics literature on audit has studied the link between greater market concentration and higher audit fees, but this does not help us very much in the present circumstances, where the primary concerns are not to do with high prices, or even exclusionary conduct, but with limited choice and sub-optimal quality. Where does the solution lie?
Penalties for poor-quality service
In public services markets (health, education) there is a high degree of regulatory supervision of quality—such as barring doctors who are found to be negligent, and awarding damages to patients harmed by negligence—even when the main providers are state-owned and have no incentive to chase profits at the expense of quality. In 2017, the UK National Health Service (NHS) estimated that the total liability for outstanding medical negligence cases could be as much as £56.1bn, and the £1.5bn annual NHS payout to settle claims is expected to double by 2023.[24] In audit, the strength of regulatory supervision by the FRC is subject to an independent review following concerns that it lacks adequate powers to intervene in the market.[25]
However, the FRC has recently been levying higher fines for audit errors. It fined PwC £6.5m regarding failed UK retailer, BHS;[26] £5.1m for its auditing of accountancy group, RSM Tenon (also, ironically, an auditor);[27] and £5m in relation to the property company, Connaught.[28] The other Big Four firms have also faced heavy fines, in both the UK and USA: £1.8m for EY’s auditing of Tech Data;[29] £4.8m for KPMG’s work on Miller Energy;[30] and £4m for Deloitte relating to the audit of Aero Inventory.[31] The FRC is also fining audit partners whom it finds to be responsible for misconduct—for example, the lead partner for BHS has been fined £325k and banned from working as an auditor for 15 years.[32] These FRC penalties are, however, minor relative to the £38m audit-related settlement reached by the UK’s largest pension scheme, USS, with PwC Brazil as part of a class action lawsuit against troubled oil giant, Petrobras.[33] But note that the FRC has this month implemented an increase in fines to £10m or more for ‘seriously poor audit work by a Big 4 firm’, following an independent review in 2017 of FRC sanctions.[34]
Are audit fines providing optimal enforcement?
From an economics perspective, if the deterrence effect of penalties is sufficiently severe, firms that might otherwise chase market share by cutting prices and their costs for a given audit will be deterred from cutting quality. In other words, when deterrence is weak, there is an opportunity for rent-seeking by firms that cut quality on unobservable dimensions. Although it might be argued that the cost to an accountant’s reputation is great enough to give the right incentives, this point seems difficult to sustain in light of the continued flourishing of firms that have had quite major hits to their professional reputations.
How large would audit fines need to be in order to deter bad audit? This article cannot provide the answer, but it may be instructive to look at a comparison between audit fines and cartel fines (in the EU). The latter are set based on the European Commission’s criteria. As the Commission explains:
The Commission’s policy with regards to competition law infringements is one of prevention … [fines] are ultimately aimed at prevention, and must hence fulfil two objectives: to punish and to deter. Breaking the competition rules is profitable if it goes unpunished – that is why companies do it.[35]
European Commission cartel fines are set based on the gravity and the duration of a competition infringement, and are capped at a maximum of 10% of a company’s total turnover. The 10% turnover ceiling for fines is engaged only when a cartel fine based on the usual criteria would otherwise be set at more than 10% of turnover.
Cartel fines are large compared with audit fines, as Tables 1 and 2 illustrate. Looking at FRC audit fines in the cases mentioned above, the average fine is 0.016% of a Big Four firm’s annual global turnover, as shown in Table 1. The final column of Table 1 indicates that increasing this percentage to 0.5% would lead to fines of a much greater order of magnitude. This is purely illustrative; it is not a recommendation as to the optimal size of audit fines.
Source: FRC and the audit firms’ annual reports for fiscal year 2017.
How do cartel fines compare? Weighted by the number of fines falling into each percentage bracket of turnover, the average European Commission cartel fine is 2.40% of turnover. This means that cartel fines expressed as a percentage of global turnover are about 150 times larger (2.40% divided by 0.016%) than FRC audit fines measured in the same way. Table 2 shows the calculation of the weighted average European Commission cartel fine.[36]
Table 2 European Commission weighted average cartel fines as a percentage of a company’s global turnover
Source: European Commission cartel statistics, last updated 21 March 2018.
It might be argued that increased deterrence for poor audit would come at the cost of competition, such as financial penalties leading to market exit and a ‘Big Three’, or hiking the barriers to entry for non-Big Four audit firms. Likewise, the Commission does not wish to fine a cartel with penalties that are so high that the consequence would be a reduction in the number of market competitors (or else the competition remedy would be self-defeating). Hence the scaling of cartel fines to turnover, and the ‘inability to pay’ test, whereby the Commission can reduce the scale of fines where it is shown that they pose a serious threat to the economic viability of the undertaking concerned. Scaling audit fines to audit firm turnover makes it unlikely that such penalties would deter entry or cause the market exit of one of the Big Four. The cartel fines policy therefore has useful principles, albeit it does not indicate the right order of magnitude for audit fines.
Fines set as a percentage of turnover would of course decline if measured against a smaller metric for revenue. As a hypothetical exercise, taking Big Four audit-only revenues as the denominator, the FRC fines mentioned previously would be on average 0.039% of the firms’ global audit-only revenues. In this scenario cartel fines at 2.40% of global turnover would be about 60 times greater than the FRC recent audit fines (2.40% divided by 0.039%), and a hypothetical fine of 0.5% of audit fines would amount to between £45m and £60m. The latter figures are much closer to the penalties proposed in last year’s independent review of FRC sanctions—i.e. ‘£10 million or more (before any discount)’. Note also that the independent review recommended that ‘the figure could be well above [£10m] if dishonesty or conscious wrongdoing were involved.’[37]
Evidence on the deterrence effect of cartel fines can be found in the economics literature. Professor Stephen Davies at the ESRC Centre for Competition Policy estimates that cartel deterrence is highly effective:
On the most conservative of our estimates, more than half of all potential cartel harm never occurs, because it is deterred. This is very much a lower bound, and the proportion could be as high as 90%.[38]
Similar research would be required to understand the effects of a different penalty regime for poor audit.
Break-up or shake-up?
There is little doubt that a new CMA investigation would consider a break-up remedy. However, no matter what the divestments and structural changes, the inherent tension within the industry’s ‘client pays’ business model is likely to remain—that is, an auditor’s basic conflict between serving the paying client and serving the greater good.
If it were to address that conflict, the CMA would need to look into penalties and deterrence, as well as studying the effects of a break-up remedy. It is not realistic to expect the CMA to be able to fix every major issue in the market by achieving the goal of reduced concentration in FTSE 350 audit.
The quality of audit might be improved with a more disaggregated market, but this link is not certain. Moreover, it is possible that greater deterrence for bad audit would lead to an organic change in market structure: the Big Four have expertise in advising clients as to when a substantial divestment or restructuring might increase shareholder value. It seems possible that, in a world of greater deterrence, the accounting firms might look inwards using this expertise and shake up the market structure themselves.
Possibly the Big Four firms are already thinking along these lines. According to a letter from the two MPs who led the parliamentary review on Carillion, voluntary break-up scenarios are now under active consideration:
Since our report was published, Bill Michael, Chairman KPMG UK, said his firm had been thinking about break-up scenarios ‘for some time’ as the current business model of the Big Four is ‘unsustainable’. Mr Michael is quoted as saying:
‘The profession, like it or not, is an oligopoly. You can’t be all things to all men and women forever. We have to reduce the level of conflicts and demonstrate why they are manageable and why the public and all stakeholders should trust us.’
Other Big four firms have reportedly begun making preparations for a break-up.[39]
Finally, the example of cartel fines shows that they are of a different scale to audit fines, raising the question as to whether fines should be reconsidered in the audit market. Penalties for anticompetitive conduct are used for prevention, not retribution. An audit firm with consistent high quality would have a minimal incidence of fines, which would place the high-quality firm at a competitive advantage to an audit firm with lower quality.[40] If audit quality became high across the market, no firm would be faced with very substantial financial penalties, and investor perceptions as to the value of statutory audit might be restored. In summary: prevention is better than cure.
[23] Peter Kyle, Member of the Business, Energy and Industrial Strategy Committee, speaking at the pre-appointment hearing with the Government’s preferred candidate for Chair of the Competition and Markets Authority, HC 985, 24 April 2018. See Transcript of oral evidence, Question 34, p. 19.
[36] The European Commission statistics provide the percentages of fines imposed on undertakings per cartel infringement. Certain cases may comprise several infringements for which multiple counting of undertakings is considered.
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Your inner Chimp can be your best friend or your worst enemy...this is the Chimp Paradox
Do you sabotage your own happiness and success? Are you struggling to make sense of yourself? Do your emotions sometimes dictate your life?
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The Chimp Paradox contains an incredibly powerful mind management model that can help you be happier and healthier, increase your confidence, and become a more successful person. This book will help you to:
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Dr. Peters explains the struggle that takes place within your mind and then shows you how to apply this understanding. Once you're armed with this new knowledge, you will be able to utilize your chimp for good, rather than letting your chimp run rampant with its own agenda. |
75 Ill. App.2d 144 (1966)
220 N.E.2d 590
Decatur and Macon County Hospital Association, a Corporation Not For Profit of Illinois, for the Use of Niagara Fire Insurance Company, Phoenix Assurance Company, Standard Fire Insurance Company, Rochester American Insurance Company, American Insurance Company, United States Fire Insurance Company, Hartford Fire Insurance Company, and Merchants Fire Assurance Corporation, Plaintiff-Appellee,
v.
Erie City Iron Works, a Foreign Corporation, T.A. Brinkoetter & Sons, Inc., a Foreign Corporation, and Illinois Power Company, an Illinois Corporation, Defendants, Erie City Iron Works, a Foreign Corporation, Defendant-Appellant.
Gen. No. 10,679.
Illinois Appellate Court Fourth District.
September 26, 1966.
Rehearing denied October 24, 1966.
*145 *146 Earl S. Hodges, of Springfield, and Greanias & Owen, of Decatur (Marshall A. Susler, of counsel), for appellant.
Giffin, Winning, Lindner & Newkirk, of Springfield (James M. Drake, of counsel), for appellee.
TRAPP, P.J.
Defendant Erie City Iron Works, hereinafter designated Erie, appeals from a judgment in the sum of $30,818.50 entered in favor of the plaintiff upon the verdict of a jury against Erie and T.A. Brinkoetter & Sons, Inc. Other disposition has been made as to the case against the latter and we consider only the appeal of Erie.
Plaintiff's action was for property damage in the approximate amount of the judgment incurred as the result of the explosion of a gas-fired boiler manufactured by Erie and installed by Brinkoetter. At the time of the explosion installation had just been completed and was at the stage of the initial start-up and adjustment of the boiler. Title to it had not yet passed to the plaintiff.
The defendant's theory is that defendant was not guilty of the negligence that was the proximate cause of plaintiff's damages; that the court should have directed a verdict in favor of this defendant, or granted defendant's post-trial motion for judgment notwithstanding the verdict of the jury or, in the alternative, should have granted defendant a new trial of the issues, because of error committed by the court in submitting, to the jury, both Count I and Count II of plaintiff's complaint, which respectively were predicated upon a res ipsa loquitur theory and specific negligence theory; that there was error by the court in denying defendant's motion for mistrial because of prejudicial conduct of counsel; that conduct of *147 a juror was prejudicial to defendant; and that there was error by the court in giving certain instructions to the jury; and other errors hereinafter discussed.
Plaintiff purchased the boiler as a "package" boiler fabricated by Erie at its plant and shipped assembled for installation as a complete unit with automatic firing controls built on.
The fire control unit and the main motorized valve were not manufactured by Erie but were purchased by it and affixed to the fabricated boiler. The Brinkoetter contract called for it to install the boiler and connect it to the line bringing gas into the building.
In making the installation, Brinkoetter did not install what has been called a "dirt leg," i.e., a trap consisting of a length of pipe extending beyond the point where a vertical gas line is turned so that it travels horizontally. Its function is to catch condensed moisture and debris in the gas line. Plaintiff had retained consulting engineers to design and supervise installation of the boiler. The schematic drawing provided by the engineer did not show a "dirt leg." The latter testified that the contractor should install a "dirt leg" whether drawn in the plans or not. Officers of Brinkoetter say that it puts in dirt legs when the plans call for them, otherwise it does not.
Neither the fabricated boiler nor the connecting line, as installed, included a "strainer," which is described as a distinctive appearing section of pipe containing a screen, the function of which is to catch debris which might be carried through the line by the flow of gas. When used, it is installed in the line ahead of the valves and controls. A brochure of the valve manufacturer recommended that a strainer be placed ahead of the main valve. Such a strainer was not included in the unit fabricated by Erie. The consulting engineer's schematic drawing did not include a strainer. He testified that he would have included it if he had known that a strainer was recommended. An officer of Brinkoetter testified that he had never heard *148 of a strainer in a gas line. In behalf of the latter, its foreman and employes testified that as the gas line was being installed, steps were taken to knock loose the scale and clean the connecting pipe. It appears that the installation was nearly completed when the contractor was advised by the gas company foreman that it would be necessary to install a regulator, i.e., a device which lowered the pressure from the 35-pound pressure in the main to some 10 pounds as specified by the boiler. A used regulator was available at the hospital and was installed. At first it did not function, but after some adjustment was observed to be reducing the pressure. It was not tested after the explosion. In installing the regulator at this time, it was necessary to cut the gas line with a torch and weld on a section of pipe. It does not appear what, if anything, was done to inspect for and remove debris in the pipe following this operation. There is some conflict in the evidence as to whether or not welding slag would enter the pipe by reason of this work.
Under the terms of its contract with Erie, plaintiff elected to have the services of a start-up engineer. Upon notification of the completion of the installation such engineer, one Enders, was sent by Erie. The explosion in issue occurred at 11:40 a.m. on Thursday, September 25, 1958. In summary, it appears that Enders had arrived on the preceding Tuesday, that the boiler was started up and fired for some 20 hours and then shut down, and that on the morning of the 25th it had been started up and fired for some 2 hours preceding the explosion. Enders died following the explosion, apparently as the result of injuries sustained.
With regard to the things done during this period, one Binns, a member of the hospital maintenance staff, testified that Enders started the boiler operation, handled the controls and made adjustments, and that immediately prior to the explosion Enders was making an adjustment of the water level in the boiler. Charles Fearn, foreman *149 of the gas distribution crew of the utility company which was working on the exterior gas line, testified that he had been in the boiler room during the morning and Enders had told him that the boiler was on low fire or "no load" firing, and that he was going to test the boiler on high fire, asking Fearn to time the meter outside so that there could be a measurement of the cubic feet of gas entering the boiler on high fire. No specific arrangement was made as to when this would be done.
Following the explosion, a State boiler inspector, and representatives of the interested parties, together with engineers and experts retained by them, assembled at the scene to examine the boiler which had been kept undisturbed. Several of them testified that they had noticed the absence of the dirt leg and the screen in the gas line connected to the boiler. The main valve was examined as to its external indicator and the testimony varies from the statement that it was apparently closed, through slightly open to one-third open. The boiler inspector testified that he assumed that it was open. It does not appear that any organized procedure was followed so that each expert present observed all of the matters testified to.
The main valve was then disassembled. Most witnesses testified to observing some scale and several pieces of welding slag on both the upstream and downstream sides of the valve.
There is testimony that upon examination of the several parts of the valve, a resilient neoprene seal was observed to be indented and that the stainless steel seat of the valve was scored to a depth of 1/16th of an inch or so, the width of the indentation being that of a blade of a table knife. There is other testimony that the seat bore only normal scratches. It does not appear that tests were made to determine whether the indentations on the neoprene seal coincided with the scoring of the valve seat. At the trial the neoprene seal no longer bore any indentation. *150 This was explained as being due to the resilient nature of the substance. The steel valve seat was not produced at the trial.
The consensus of the testimony is that there was a gas explosion followed by an explosion of the boiler itself. The opinion testimony is that the first explosion resulted from the ignition of a surplus of gas within the combustion chamber, which gas was somehow ignited. Paul Wilson, an employe of Erie in charge of their service department, testified that he did not believe it possible to find the actual cause of the majority of explosion cases, and George Harper, a professor of engineering at the University of Illinois, testified that in such an explosion things are so disrupted that it cannot be ascertained with certainty what happened, but that it was necessary to draw deductions.
From the record it appears that a variety of factors inducing the explosion may have existed. There is, of course, the contradictory nature of the testimony as to whether or not the motorized main valve was closed or open, whether or not slag from welding had lodged in the main valve so that it was not completely closed, and whether such slag would be sufficient to hold the valve open with the pressures concerned without distorting the valve stem, which apparently was in normal condition.
There is testimony by Ted Brinkoetter that the control system, upon being tested, did not always work, but there is also testimony that it functioned correctly upon tests. Harry Reynolds, an investigating engineer retained by the plaintiff, testified that it would take a very small amount of gas to cause an explosion in this boiler, and that it was particularly hazardous to operate the boiler on a "no load" basis as the mixture of air and gas gets out of balance and becomes explosive. He also testified that upon initial examination, the oil burning switch was on instead of the gas burning switch. A witness, testifying in behalf of Brinkoetter, stated that shortly before the explosion, *151 Enders flipped a switch and that the flame in the boiler went out and did not come on again.
It is one of defendant's arguments that by this contract it was to furnish a package boiler but had no responsibility for its installation. This position was taken in its first motion to the complaint and is argued here.
The nature of defendant's disclaimer seems to be based upon its Exhibit #1 contained in a foreword to the instruction manual which Erie shipped with the boiler. A relevant part includes the following:
"When the service of an Erie City Iron Works Engineer is provided for the customer, it is for the purpose of aiding in the training of the customer's personnel and not to replace them or assume any of their duties. It should be understood that the responsibility for operation rests solely with the customer's operators and the Erie City Iron Works assumes no responsibility for the customer's operators' failure to properly perform their respective duties, and the presence of an Erie City Iron Works Engineer at the customer's plant in no way relieves the customer's personnel of any of their responsibilities."
The following also appears in slightly varying form in several places in the contract for the purchase of the boiler:
"With respect to all preliminary operations, initial start-up, demonstration of capacity and performance guarantees, representatives of the Company are authorized only to advise and consult with the Purchaser or its representatives and no representative of the Company is licensed to operate the equipment. In the event the Purchaser shall operate the equipment specified hereunder prior to final acceptance, the Purchaser shall indemnify and save harmless the Company against any loss or expense and against any liability imposed upon the Company, resulting *152 from the operation of such equipment by the Purchaser prior to final acceptance, except any such loss, expense or liability for injury or damage resulting from the negligent acts or omissions of the Company or its agents or employees." (Emphasis supplied).
It appears from the testimony that the package boiler is not operational upon delivery but requires adjustment to make it perform properly. Paul Wilson, who is in charge of field service for defendant, testified that the linkage of the butterfly valve regulating the ratio of air and gas must be adjusted and that the damper linkage must be "positioned." He testified that the service engineer never operates the boiler but that it is the obligation of the purchaser to make such adjustments according to the engineer's instructions. He testified that it was the service engineer's duty to make a visual check of the gas line installed, check the controls and firing equipment, consult and assist placing the boiler in service, instruct in operating the boiler and its controls and assist in making the final adjustments.
Brewster, a witness for Brinkoetter, testified that Enders examined the pipeline but made no suggestions for changes in the work as installed, and the record is that Enders did, in fact, start-up and fire the boiler, make adjustments, and made or had arranged to make the tests, including the testing of its capacity on the high fire. Binns, an employe of the hospital, testified that no one other than Enders handled or adjusted the controls. The manual submitted by Erie contains a section A designated "Preparing the boiler for service Inspection of unit." Section A-1 states that prior to placing equipment in service a complete inspection should be made to determine its condition and continues:
"In case of newly constructed power equipment, this inspection should insure that the unit has been correctly completed."
*153 Section A-2 is as follows:
"Responsibility for the completion of construction normally rests with the customer's construction engineer working in conjunction with the manufacturer's erection or service engineer. At completion of construction work, an inspection should be made in the presence of the customer's construction engineer, operating engineer, the construction superintendent and the manufacturer's engineer (if one is present) and agreement reached that the equipment is in a satisfactory condition for placing into service."
There is no evidence that such inspection or agreement was reached or called for by defendant's service engineer.
As to the contention that by contract Erie had no responsibility, claimed under its Exhibit #1, the "foreword" to the instruction manual and the several provisions set out in the contract should not control under these circumstances. The effect of these documents might be that Erie could not be required to perform the tests and effect the start-up of the boiler, but they should not control liability where under the evidence it might be reasonable to conclude that they did, in fact, undertake and perform the work. The contract provision quoted does not attempt to exclude negligence of Erie employes.
Erie discusses Count I of the complaint as involving the principles of res ipsa loquitur under a pleading of general negligence. These principles are thoroughly discussed in Metz v. Central Illinois Electric & Gas Co., 32 Ill.2d 446, 207 NE2d 305, and need not be reiterated.
[1] Erie urges that the inference of negligence under Count I should not be allowed because the boiler was not under its exclusive control. The defendant points out that the evidence discloses that Enders, Brewster, an employe of Brinkoetter, Binns, an employe of the hospital, and Robert Brinkoetter were all present at the time of the explosion. The evidence has been examined to determine *154 what, if anything, these individuals were doing to exercise control of the unit. We cannot say that it is contrary to the manifest weight of the evidence for the jury to conclude that Erie's man Enders was, in fact, in control of the proceedings incident to the start-up and testing of the boiler. There is no evidence that any person other than Enders participated in any phase of the work.
In May v. Columbian Rope Co., 40 Ill. App.2d 264, 189 NE2d 394, the complaint alleged the purchase and delivery of a new rope which broke shortly after placing the rope into use. There was judgment n.o.v. entered by the trial court. The Appellate Court reversed, holding that the inference of negligence under the theory of res ipsa loquitur was properly applicable. As to that defendant's contention that it was not in control of the rope at the time of the injury, the court said:
"Decisions from other states and recent cases here reject this inflexible application of a rule of control and hold that a defendant in a res ipsa loquitur case cannot automatically defeat an allegation of negligence with a bare showing that, before harm struck, it had parted with control of the harmful instrumentality. (Prosser, Torts 206 (2d ed 1955).)
"The demonstrable trend of these authorities is to determine from the nature of the defective instrumentality and the surrounding circumstances whether the inference of the defendant's negligence is strong enough to survive the fact that, between the defendant's control and the plaintiff's injury, another possession intervened."
The court continued to say that it was for the determination of the jury as to whether the permissive inference of negligence arising from the facts was to prevail over defendant's countervailing proof of due care.
As stated in Prosser, Law of Torts, 2d ed 1955, p 206, chap 7, § 42, the word "control" may be the wrong word. It is said:
*155 "Some courts have said that it is enough that the defendant was in exclusive control at the time of the indicated negligence. It would be far better, and much confusion would be avoided, if the idea of `control' were discarded altogether, and we were to say merely that the apparent cause of the accident must be such that the defendant would be responsible for any negligence connected with it."
In Schroeder v. City & County Sav. Bank of Albany, 293 NY 370, 57 NE2d 57, the defendants were several contractors and the owner of a building under repair. The court noted:
"It is not necessary for the applicability of the res ipsa loquitur doctrine that there be but a single person in control of that which caused the damage."
Amongst other cases defendant relies upon Kirchner v. Kuhlman, 334 Ill. App. 339, 79 NE2d 628. There defendant's employes were working on plaintiff's premises but we find no evidence that these defendants had control of the trash container belonging to the plaintiff in which the fire started. Again, in Krump v. Highlander Ice Cream Co., 30 Ill. App.2d 103, 173 NE2d 822, the collision of two automobiles caused one of them to strike and damage plaintiff's building. While the court said that the doctrine of res ipsa loquitur did not apply, it did hold that there was a presumption of negligence where an accident occurred which would not ordinarily occur if due care had been taken, and that it was proper to call upon the defendants to exculpate themselves. The distinction between this conclusion and the theory of res ipsa loquitur appears slight.
[2] Defendant argues that Count I of the complaint alleged general negligence stating a cause of action upon the theory of res ipsa loquitur, while Count II alleges certain acts of specific negligence, and that under the authorities in this State the inference of negligence which *156 arises under res ipsa loquitur, "vanishes" upon the introduction of evidence of specific negligence. Amongst the authorities cited are Bollenbach v. Bloomenthal, 341 Ill. 539, 173 NE 670. This rule has been categorically overruled by our Supreme Court in Metz v. Central Illinois Electric & Gas Co., 32 Ill.2d 446, 207 NE2d 305. In that case the complaint charged general negligence in one count employing the theory of res ipsa loquitur, and in a second count alleged specific negligence. At the close of the evidence plaintiff was required to, or did elect, to rely upon the charge of negligence and the theory of res ipsa loquitur. The verdict for the plaintiff was reversed in the Appellate Court on the theory that res ipsa loquitur did not apply as other parties had access to the area of the gas main. In reversing the Appellate Court, the Supreme Court remarked upon the conflict amongst the Illinois decisions. We may note that many of these decisions are in broad language open to a variety of interpretations, and frequently they do not indicate the reason for the decision. In Metz the Supreme Court concluded that the more studied, more just view is that the inference of negligence does not vanish when contrary evidence appears, but that it remains to be considered and weighed by the jury against the direct evidence offered by the party charged, citing Cobb v. Marshall Field & Co., 22 Ill. App.2d 143, 159 NE2d 520; Illinois Pattern Jury Instruction, 22.01 with comment on pages 128, 129; Prosser, 20 Minn L Rev, 241. See also O'Hara v. Central Illinois Light Co., 319 Ill. App. 336, 49 NE2d 274; May v. Columbian Rope Co., 40 Ill. App.2d 264, 189 NE2d 394.
[3] Defendant's contention that plaintiff should have been required to elect as between the counts is controlled by the rule of Metz. Defendant's authorities are Wm. Wrigley, Jr. Co. v. Standard Roofing Co., 325 Ill. App. 210, 59 NE2d 510; and Simmons v. South Shore Hospital, 340 Ill. App. 153, 91 NE2d 135. In the former case the Appellate Court undertook to specify what may be described *157 as the requirements that plaintiff elect between the general negligence count and the count for specific negligence. The only cited authority for such procedure was Bollenbach v. Bloomenthal and its rule that the inference of negligence vanished upon the introduction of evidence of specific negligence. By reason of the Metz decision, this reason for such rule no longer exists. Simmons v. South Shore Hospital, as well as Jackson v. 919 Corp., 344 Ill. App. 519, 101 NE2d 594, simply relied upon the rule of Wrigley as authority without discussing it.
There is, in fact, persuasive opinion contrary to the contention of Erie regarding the theory of election in Erckman v. Northern Illinois Gas Co., 61 Ill. App.2d 137, 210 NE2d 42. There premises were damaged by an explosion of gas leaking from the company lines. The complaint alleged only specific negligence and there was some evidence of a failure of periodic inspection. The trial court gave an instruction authorizing the jury to apply, or employ, the inference of negligence under res ipsa loquitur. The Appellate Court reversed since there was no pleading of general negligence, but stated that upon a new trial the complaint should be amended to include such an allegation. The court there said:
"An inference of general negligence arising from the doctrine of res ipsa loquitur is not necessarily inconsistent with proof of specific negligence. To hold that proof of specific negligence precludes the application of the res ipsa doctrine could lead to the absurd result of weak proof of specific negligence voiding a strong inference of general negligence.... If there is an inference of general negligence and proof of specific negligence, but reasonable men may differ as to the effect of this evidence, it should then be for a jury to determine under which theory, if any, the plaintiff should prevail. McCormick v. Kopmann, 23 Ill.2d 189, 205, 161 NE2d 720 (3rd Dist 1959)."
*158 [4] The Illinois courts recognize that the doctrine of res ipsa loquitur is but one form of circumstantial evidence. May v. Columbian Rope Co., 40 Ill. App.2d 264, 189 NE2d 394.
It has been suggested that the doctrine that requires election assumes that the inference arising through res ipsa loquitur must be an alternative to direct proof rather than a type of circumstantial evidence to be weighed with other evidence, and it has been criticised as an assumption that the pleader must be totally ignorant of the facts. 2 ALR3d 1335, at 1340. There is reason in the hypothesis that there should not be a penalty imposed upon the pleader for placing before the court all facts known to him. 27 Fordham L Rev, 411-415; Foster v. Union Starch & Refining Co., 11 Ill. App.2d 346, 137 NE2d 499. This is particularly true when an allegation notifies the defendant of the intent to rely upon the inference of negligence arising under the doctrine of res ipsa loquitur. It is the policy under the rule of Metz v. Central Illinois Electric & Gas Co., 32 Ill.2d 446, 207 NE2d 305, that once the inference of negligence arises through allegations of general negligence, it remains for the consideration of the jury, unless and until the precise cause of the injury is established. 27 Fordham L Rev 411. In Prosser, Law of Torts, 2d ed, chap 7, § 43, p 214, it is suggested:
"It is quite generally agreed that the introduction of evidence which does not purport to furnish a complete explanation of the occurrence does not deprive the plaintiff of res ipsa loquitur."
In Cassady v. Old Colony St. Ry. Co., 184 Mass. 156, 68 NE 10, at p 12, the court said:
"The defendant also contends that, even if originally the doctrine would have been applicable, the plaintiff had lost or waived her rights under that doctrine, because, instead of resting her case solely upon it, she undertook to go further, and show particularly *159 the cause of the accident. This position is not tenable. It is true that, where the evidence shows the precise cause of the accident, (citing authorities), there is, of course, no room for the application of the doctrine of presumption. The real cause being shown, there is no occasion to inquire as to what the presumption would have been as to it if it had not been shown. But if, at the close of the evidence, the cause does not clearly appear, or if there is a dispute as to what it is, then it is open to the plaintiff to argue upon the whole evidence, and the jury are justified in relying upon presumptions, unless they are satisfied that the cause has been shown to be inconsistent with it. An unsuccessful attempt to prove by direct evidence the precise cause does not estop the plaintiff from relying upon the presumptions applicable to it."
We believe that this position was approached in Krueger v. Richardson, 326 Ill. App. 205, 61 NE2d 399, when the court noted that the plaintiff was not required to prove the specific acts of negligence as alleged, but they had a right to rely upon the proof and its reasonable inferences to establish a prima facie case of general negligence.
In this case it seems proper to say that reasonable men might differ as to the effect of the evidence heard by the jury. Expert witnesses would not even undertake to announce an hypothesis, but rather advised of the virtual impossibility of reaching a specific determination of what caused the explosion. This situation here appears to be precisely that contemplated in the language of Erckman v. Northern Illinois Gas Company.
[5] In its reply brief Erie contends that the doctrine cannot be followed because there are multiple defendants. No Illinois cases seem applicable as precedent. In Schroeder v. City & County Sav. Bank of Albany, 293 NY 370, 57 NE2d 57, it was held error to dismiss a complaint seeking to apply res ipsa loquitur as against three defendants. *160 See also Burr v. Sherwin-Williams Co. (Cal App), 258 P.2d 58, 38 ALR2d 905 et seq. Again in Zichler v. St. Louis Public Service Co., 332 Mo 902, 59 S.W.2d 654, general negligence was pleaded against the service company while specific negligence was pleaded as to another defendant who was found not guilty by the jury. It was contended that it was improper to permit the res ipsa loquitur inference to be applied to one joint tort feasor, but not the other. Pointing out that the rule was one of evidence rather than pleading, the court said:
"A plaintiff should not be compelled to confine his action to one joint-feasor only in order to be accorded the rights which the law gives to him."
It being the policy under the rule of Metz that the inference of negligence is to be weighed by the jury with other evidence, we see no reason why the benefit of such rule should be denied to the plaintiff where under the events at issue, more than one party may be the source of injury to the plaintiff for otherwise he would be limited in the use of, or be completely denied the benefit of the rule. In Metz the Supreme Court said that whether the doctrine applies in a given case is a question of law for the trial court. We believe that these conclusions dispose of the contentions of Erie that the court erred in refusing to strike par 8 to Count I.
Defendant contends that the case must be remanded for error in the giving of instructions. His objection to plaintiff's instruction #20 is that it permits the jury to consider the case upon the theory of res ipsa loquitur, as well as upon the allegations of specific negligence. The matters hereinabove discussed dispose of this contention.
[6] There is objection to Brinkoetter's instruction #6 which may be summarized as an issues instruction relating to negligence alleged as to Erie and as to the defendant Brinkoetter. It is contended that as to Erie there is no evidence in the record as to certain matters *161 stated in the instruction to be alleged in the complaint. The Abstract discloses that at the conference on instructions Erie simply made the objection that the evidence did not support all of the charges. This does not meet the rule that specific objections to instructions must be made at the conference on instructions. Vasic v. Chicago Transit Authority, 33 Ill. App.2d 11, 180 NE2d 347. The court's comment indicates that he believed that those matters not supported by the evidence had been omitted from the instruction. Under such circumstances we do not believe that there is reversible error.
[7] Erie urges that the cause must be reversed and remanded by reason of the fact that a juror on voir dire indicated that he was not interested in any lawsuits then pending in court, but that subsequent to the trial, counsel discovered that he had been defendant in a lawsuit and was, at the time of trial, a plaintiff in a pending cause. Erie does not contend that it was, in fact, prejudiced by the juror sitting upon the panel, but says that the prejudicial effect cannot be calculated. It indicates that it could have challenged the juror, though it is not claimed that it would have done so. In Department of Public Works & Buildings v. Christensen, 25 Ill.2d 273, 184 NE2d 884, it was alleged that the party would not have accepted the juror if a true answer had been given. The Supreme Court there held that the motion for a new trial would be denied unless it was shown not only that the juror answered falsely, but also that prejudice resulted. Erie cites the case of People v. Ortiz, 320 Ill. 205, 150 NE 708, which may be distinguished because in that case the juror had actually expressed hostility to the defendant which he had concealed.
[8] Erie urges that the judgment must be reversed because of a reference to insurance introduced during cross-examination in behalf of the defendant Brinkoetter. One George Harper testified in behalf of the plaintiff as an expert witness who had examined the boiler following *162 the explosion. It appears that he had originally been requested to make the examination by a representative of the company insuring Erie. The name of the insurance company was given in answer to a question to whom he had delivered his report. The trial court sustained an objection to a question as to what party was covered and an objection as to whether the insurance company represented Erie. The trial court, while indicating disapproval of counsel's action, denied the motion for a mistrial.
It is clear that plaintiff did not, in any way, precipitate this issue. Under the circumstances of this case, the proceedings clearly indicated to the jury that certain insurance companies were to be the beneficiaries of a judgment for plaintiff. This fact would seem to indicate little probability of prejudice as between insurance companies upon the issue of liability. Edwards v. Hill-Thomas Lime Co., 378 Ill. 180, 37 NE2d 801.
Upon the possibility of prejudice regarding the issue of damages, the amount of the verdict is slightly less than the amount paid by plaintiff to Erie for the boiler. Insofar as counsel may have attempted to create prejudice as between the parties defendant, the verdict of the jury is joint and they seem to make no distinction. Under the circumstances of this case, we conclude that there was no abuse of discretion by the trial court in refusing to grant a mistrial. Isenhart v. Seibert, 6 Ill. App.2d 220, 127 NE2d 469.
[9] Upon consideration of the issues of law, we conclude that the trial court did not err in refusing to direct a verdict or enter a judgment n.o.v. upon the several motions made by Erie, and that, from an examination of the evidence, the verdict of the jury is not contrary to the manifest weights of the evidence.
Taken with the case was plaintiff's motion to dismiss as a "use plaintiff" the Niagara Fire Insurance Company. The effect of such dismissal is to reduce the amount of *163 the judgment in the sum of $4,873.05. The motion is allowed and the judgment ordered reduced in said amount.
The judgment of the trial court is affirmed, but the cause is remanded with directions to enter judgment in the amount due by reason of the dismissal of the party plaintiff pursuant to motion.
Affirmed as modified.
SMITH and CRAVEN, JJ., concur.
|
Comparison of patient satisfaction with acrylic and flexible partial dentures.
Restoration of partial edentulous mouth may be done using a variety of treatment options. Removable partial denture (RPD) is commonly used because of its availability. RPDs from flexible resins unlike those from acrylic engage hard and soft tissue undercuts and feel more comfortable in the mouth. The aim of the study was to compare satisfaction with dentures made from these two types of materials. It was a quasi-experimental study among thirty patients at the Prosthetics Clinic, University College Hospital, Ibadan. Patients aged 16 years or more, requiring RPDs with one to three missing teeth in the anterior region of either the upper or lower arch participated. A modified semi-structured interviewer-administered questionnaire was used to collect data on sociodemographics and oral health variables. The level of satisfaction was assessed using a visual analogue scale. Data were analysed using descriptive and multivariate statistics at a significance level of P < 0.05. The participants' ages ranged between 16 and 51 years, mean age was 33.8 ± 10.01 years. Male: female ratio was 1:1 and mean duration of edentulousness was 11.37 ± 10.52 years (median - 9.50). Most 28 (93.3%) subjects delayed replacement of their missing teeth; reasons were indifference 13 (43.4%), financial constraint 10 (33.3%), ignorance 4 (13.3%) and fear of aspiration 1 (3.3%). Overall, 21 (70.0%) participants were more satisfied with the flexible dentures, 6 (20.0%) with acrylic dentures while 3 (10.0%) were equally satisfied with both types of dentures (P = 0.04). Subjects were more satisfied with the flexible RPD than the acrylic resin RPD. |
ZURB Tavern - jacobwg
http://zurb.com/tavern
======
pepsi
By the name, I thought that this was going to be a MUD.
|
Rigid stretchers for transporting injured patients are well known. Certain known rigid stretchers are partially collapsible. These stretchers include one or more rigid support panels or beams. Because of the rigid panels or beams, these stretchers can be relatively heavy and cumbersome when handled by emergency personnel during rescue operations, and these stretchers can occupy a relatively significant amount of space in vehicles and other storage areas. Also, these known stretchers do not include a patient covering which aids in the protection of emergency personnel from hazardous body fluids from the patient and which guards the front of patient's body during transport.
One known rescue bag has been developed for keeping injured people warm while they are lying on stretchers. Though this rescue bag covers part of the patient's body, it is merely an accessory to a stretcher. Accordingly, one of the disadvantages of this rescue bag is that it does not function as a patient carrier. The emergency personnel must use a stretcher in conjunction with this rescue bag in order to pick-up, carry and transport an injured person to a desired location. In addition, such a rescue bag does not have medical treatment openings which provide emergency personnel with relatively quick access to select portions of the person's body, for example, to deliver essential treatments, such as IV solutions, heart defibrillation and the like.
Therefore, there is a need to overcome the foregoing disadvantages and to provide improvements to patient transporters. |
Q:
Reading a text file and moving files from it to a directory
I have a directory of images. Some of these images must be stored in a text file like 'pic1.jpg'
I need to extract this filename, pick up the matching file from the current working directory and move it to a separate folder (under the cwd).
This is the code I have so far, but I cant get the shutil operations to work. What am I doing wrong?
Current directory C:\BE
Have to move a file(s) 1,jpg, 2,jpg etc from a textfile called "filelist.txt" to C:\BE\2014-03-25_02-49-11
import os, datetime
import shutil
src = os.getcwd()
global mydir
def filecreation(content, filename):
mydir = os.path.join(os.getcwd(), datetime.datetime.now().strftime('%Y-%m-%d_%H-%M-%S'))
try:
os.makedirs(mydir)
except OSError, e:
if e.errno != 17:
raise # This was not a "directory exist" error..
with open(os.path.join(mydir, filename), 'w') as d:
d.writelines(content)
#shutil.copyfile(src,mydir)
def main():
filelist = "filelist.txt"
with open(filelist) as f:
content = f.read().splitlines()
#content = shutil.copyfile(src, mydir)
print content
print "Here we are"
#list=['1.jpg','2.jpg']
filecreation(content,"filelist.txt")
print "lets try another method"
with open('filelist.txt','w+') as list_f:
for filename in list_f:
with open(filename) as f:
content = f.read()
#content = shutil.move(src,mydir)
#content = shutil.copyfile(src,mydir)
#for line in f
print "method 2 is working so far"
if __name__ == '__main__':
main()
A:
This is what finally worked -
from shutil import copy
f = open(r'C:\Users\B\Desktop\BE Project\K\filelist.txt', 'r')
for i in f.readlines():
print i
copy(i.strip(),r"E:\Images")
f.close()
|
Q:
Are all I2C sensors interoperable?
I have a quadcopter flight controller (RTFQ Flip MWC) that supports I2C sensors for adding thing like a barometer, magnetometer, and GPS system. The officially supported sensor block (BMP180, HMC5883L on one board) is discontinued, as far as I can tell.
I have found other I2C barometer and magnetometer sensors, (BMP280, LSM303) but I am not even sure if all I2C devices of the same type are interoperable. Do they all look the same (at least interface-wise) to the flight controller?
I'm also new to I2C in general; the sensors I need come on two separate boards. Do I just stack the boards, directly connecting the I2C bus between each?
Thanks in advance,
Neil
EDIT:
I was able to find the datasheets for the discontinued and proposed sensors:
BMP180
HMC5883L
BMP280
LSM303
All are compatible with the 3.3v output of the Flip MWC, which is good.
I was quickly able to find what I believe to be the register map for the BMP180 and HMC5883L, but the table I found for the LSM303 was very confusing and I wasn't able to find one in the BMP280 datasheet.
A:
The only way to know if two IIC devices are compatible in this context is to compare their IIC interface in the two datasheets very carefully. IIC may be largely standard, but it says nothing about the payload data carried over IIC.
If a particular product becomes popular, competitors will often make theirs compatible. However, there is no guarantee that any two devices are compatible. Each could use a different format for sending the data, require different settings in different registers that are accessed differently to select features, etc.
Unless you know they are compatible, assume they are not.
|
Randomised trial comparing forced-air warming to the upper or lower body to prevent hypothermia during thoracoscopic surgery in the lateral decubitus position.
In the supine position, forced-air warming is more effective on the lower body than on the upper body to prevent intraoperative hypothermia. However, it is unknown in the lateral decubitus position. We thus compared forced-air warming on the upper and lower bodies in the lateral position. Patients (n=123) were randomised to receive forced-air warming on the upper body or lower body during thoracoscopic surgery in the lateral position. We measured the nasopharyngeal temperature at 0, 30, 60, 90, and 120 min after lateral positioning during surgery and the infrared tympanic membrane temperature at 0, 30, 60, 90, and 120 min after surgery. Patients received both upper and lower body warming at a temperature of <35.5°C. The primary outcome was the incidence of intraoperative hypothermia with a temperature of <36.0°C. Intraoperative hypothermia was less frequent with the upper body warming than with the lower body warming {21/62 vs 35/61, risk ratio [95% confidence interval (CI)] 0.6 (0.4-0.9), P=0.011}. The intraoperative temperature was higher with the upper body warming than with the lower body warming at 30 (P=0.002), 60 (P<0.001), and 90 (P<0.001) min after lateral positioning, and the postoperative temperature was higher at 0 (P<0.001) and 30 (P=0.001) min after surgery. Fewer patients received both upper and lower body warming in the upper body warming group than in the lower body warming group during surgery (1 vs 7, P=0.032). Forced-air warming was more effective on the upper body than on the lower body to prevent hypothermia during thoracoscopic surgery in the lateral decubitus position. NCT02993666. |
+ 73 = -0*m - 7*m + 5*m for m.
4
Solve 0 = -6*x - 27*p + 276, 73*p - 72*p = 4*x - 146 for x.
37
Solve 6*n + 2828 = 5*z + 3287, -2*z - n = 187 for z.
-93
Solve 14*b = -b + h - 58, 5*b + 40 = -5*h + 10 for b.
-4
Solve 9*d - 537 = 3*l, -d = 5*l - 1560 + 1511 for l.
-2
Solve 4*h + 2*t + 0*t - 4*t = -0*t - 38, -43*t = -46*t - 3 for h.
-10
Solve 5*l - 10*f - 2976 + 3071 = 0, -2*l - 3*f = -11 for l.
-5
Solve -3*s - 217*i - 22 = -215*i - 4, -10*s - 49*i - 60 = -51*i for s.
-6
Solve 18*a + 5*q = 23*a + 30, 0 = -7*a + q - 36 - 18 for a.
-8
Solve 3*i - 6*i - q = -6, 6*i + 52*q = 10*q + 38 + 94 for i.
1
Solve -3*h - 7001 + 7102 = -o, 3*o - 5*h + 194 = -89 for o.
-86
Solve 4*r = 7*r - 132, -355*q - 5*r + 98 = -351*q - 122 for q.
0
Solve -4*n - 195 = -3*n + 5*i, -129 = 697*n - 694*n + 3*i for n.
-5
Solve 5*d - 27 = 4*r, r - 10*r + 10*d - 39 = 6*d for r.
-3
Solve -3*s - 29*a + 10*a = -17*a + 28, 0 = -2*s - 5*a - 26 for s.
-8
Solve 5*g + 0*g + j - 2 = -17, -70*g + 68*g = 2*j + 6 for g.
-3
Solve -o + 4402*r + 6 = 4401*r, -30 = -5*o + 6*r for o.
6
Solve -93 = -u + 3404*t - 3402*t, 2*u + 10*t + 302 = -142 for u.
3
Solve -2*j = -9*j + 2*i + 302 - 19, 2*j + 4*i - 22*i = 168 for j.
39
Solve -5*v + 0*v - 230 = 0, -17*m = v - 3 - 2 for m.
3
Solve 35 = 3*g - k, 20 = 4*g + 133*k - 129*k for g.
10
Solve -3*r - 5*z - 4 = 0, 50891*z - 50894*z + 8 = -r + 2 for r.
-3
Solve -124*l + u = -127*l + 15, -5 = 2*l - u for l.
2
Solve -3*q - 419 = -5*x - 0*q - 352, 5*q = -126*x - 246 for x.
-1
Solve -5089032*l = 5*t - 5089027*l + 60, 2*t - 5*l + 6 - 40 - 19 = 0 for t.
-1
Solve -40*q - 42 = 3*y - 46*q, 6*y - 8*y - 4*q = -12 for y.
-4
Solve y + 3 = 8196*b - 8194*b, b - 2*y = 6 for b.
0
Solve 5*t = 154*z - 123*z - 377, -4*z - 4*t = -44 for z.
12
Solve s = -2*t - 3*t - 56, 5*t + 456*s + 274*s = -785 for t.
-11
Solve f - 2 = -2*u, -5*u = -5*f - 37080 + 37150 for f.
10
Solve 0*z + 5*t = -3*z + 50, -4*z - 3*t + 16 = -20 - 5 for z.
5
Solve -375 + 527 = 4*l - 4*y, -5*y + 190 = 5*l for l.
38
Solve 2*d - 802 = -6*t - 1180, -t - 63 = -5*d for t.
-63
Solve -2*c + 7 = i, 4*i = 4*c - 2715 + 2707 for i.
1
Solve q - 2*q + 57*o - 60*o + 91 = 70, -5*q + 40 = 2*o for q.
6
Solve 40 = -12*u - 8, 0 = 183*r - 186*r + 2*u + 2 for r.
-2
Solve -1828*z + 1812*z - 298 = -5*f, 0 = 3*f - 6 - 0 for z.
-18
Solve -2*p + 16*s - 3 + 9 = 0, 2*s = 3*p - 7*p - 2*s - 60 for p.
-13
Solve 0 = -3*f - 3*p + 9, -1916*p = 4*f - 1920*p - 20 for f.
4
Solve 4*f + 2*s + 22479 = 22575, 3*f - 21*s - 27 = 0 for f.
23
Solve n - 4 = -8*x + 5*x, 5*x - 103*n + 101*n + 5 = -3 for x.
0
Solve -13*r + 28616*n - 28612*n - 176 = 0, -43 = -298*r + 301*r - n for r.
-4
Solve 4*b + 35*m - 1229 = 0, 104*b - 212*b = -105*b + 4*m - 143 for b.
1
Solve j - 12 - 363 = 5*i - 37, j = 3*i + 202 for i.
-68
Solve -3*d + 32 = -4*r, 55*r - 26*r + 3*d = 28*r - 1 + 8 for r.
-5
Solve 12*q - 214 = 5*u, -39*u - 17 - 647 = 2*q + 814 for q.
2
Solve -6 = -h + i, -4*i = -13*h - 1982125 + 1982131 for h.
-2
Solve 0 = 123291*b - 123286*b + 7*o - 5, -4*o = 0 for b.
1
Solve -13*c = 3*d - 9*c - 9*c + 137, -d + 5*d + 186 = 5*c for d.
-49
Solve -2*i + 226 - 83 = -l, -3*i - 2*l = -163 - 62 for i.
73
Solve 0 = -3*a + d - 22, -10*a + 38 = 2*a - 102*d + 35*d for a.
-8
Solve d + 4*d = l + 32, 0 = 4*l - 26*d - 1095 + 1229 for l.
-27
Solve 64*s + 286 = -2*w - 141 + 37, 3*w = -5*s - 39 for w.
-3
Solve -86*v + 82*v + j - 6 = -0*j, -2*v = -5*j + 102 for v.
4
Solve -w + 7*i + i - 7*i = -5*w, 27*w + 2*i - 76 = 0 for w.
4
Solve 19 + 68 = 25*a - 2*p, 391*a - 387*a + 4*p - 18 = 5*p for a.
3
Solve 0 = -11*i - 44, -3*i + 0*i + 22 = 516*o - 514*o - 0*i for o.
17
Solve 0 = 3*r + 34*x - 31*x - 18 - 12, 95*r - 562 = 2*x for r.
6
Solve -5*z = v + 7, -3*z + 5*v + 110 = 2*z + 19*v + 26*v for z.
-2
Solve 3*q - 3*a = 48, -29854*q - 8*a - 29 = -29853*q for q.
11
Solve -54737*u + 54733*u = 45*s - 37, -s - 3 = 2*u for s.
1
Solve -5*u + 15*r = 19*r - 22, -7*u + 6*r = -4*u + 3*r + 3 for u.
2
Solve 2*w + 4*m = -22, 0 = -520*w + 530*w - 2*m + 88 for w.
-9
Solve 111*q + 5 = 104*q + 2*f, -3*f + 45 = 2*q for q.
3
Solve -12889 + 12916 = 12*j - 15*j, -5*j - 25 = 5*b for b.
4
Solve 0 = -5*s - 100*c + 745, -12*s - 2*c + c = -16*s + 191 for s.
49
Solve -l - 4*b = 3*l - 2128 + 2060, -10*l + 275 = 5*l + 5*b for l.
19
Solve 161 + 185 = 46*d - 7*d - 15*s + 16*s, s = 5*s + 20 for d.
9
Solve t + z = -8, 26*t + 43*z = -14*t + 39*z - 248 for t.
-6
Solve 3*p - 6*t = 66, 0 = -2*p + 2654*t - 2652*t + 34 for p.
12
Solve 67*y - 44*y - 19*y + 4*f - 76 = 0, 3*y + 4*f - 76 = 0 for y.
0
Solve -745454*h + 20 = -745459*h, -4*t = -3*h - 4 for t.
-2
Solve 5*y - 9*j = -12*j + 32, 16*j - 20*j - 4 = -5*y for y.
4
Solve 403*r - 400*r + 46 = a, -5*a = -2*r - 355 + 99 for r.
2
Solve 43*f = 45*f - 2*k - 24, f + 53 = -4*k for f.
-1
Solve 2*h - 79 = 14*z - 13 + 88, -4*h - 33 = 3*z for z.
-11
Solve -14*c + 2032*n = 2034*n + 50, 0 = -5*c - 2*n - 14 for c.
-4
Solve -x = 6*t + 13, -3*t = 5*x + 2042747 - 2042682 for t.
0
Solve -11*h = 4*i - 14*h + 70, h - 25 = 3*i for i.
-1
Solve 41*g = 53*g + 48, -3*v = -v - 5*g + 6 - 32 for v.
3
Solve -2886 = 207*w - 4*k, 28553*k - 28550*k = 4*w + 47 for w.
-14
Solve 0 = -2*n - o - 21, 960*o + 18 = -3*n + 963*o for n.
-9
Solve -3*z = 0, 1 - 37 = -5*c - 8*z - 10*z + 19 for c.
11
Solve 2*v - a + 9 + 34 = -0, 163 = 3*v + 5*a for v.
-4
Solve a + 6*y = 11*y - 45659 + 45672, 3*a - 4*y + 5 = 0 for a.
-7
Solve 245*l - 496*l = -246*l - 3*d - 18, 17 = 4*l - 5*d for l.
3
Solve 2*z - 13*v = -159, 254*z - 250*z = 4*v - 208 for z.
-47
Solve -25 = -3*b + 9737*z - 9733*z, -5*b + 3*z + 60 = 0 for b.
15
Solve 3*j + 3*d = 4 - 19, 0 = 9*j - 3*d + 93 for j.
-9
Solve -4 = 6*d - 2*x, 3*d - 1595*x - 8 = -1596*x for d.
1
Solve 2*z - 4*d - 50 = 5*z, -38*z - 355 = d - 6*d for z.
-10
Solve -35*b + 78*b = 8*j + 42*b + 7, b = 3*j + 2 for j.
-1
Solve -17*q = i - 55, 4*i + 2489 = 5*q + 2490 for i.
4
Solve -5*y - 7*p - 110 + 12 = 0, -8*y - 51 = -6*y - 9*p for y.
-21
Solve 9081 = -p - 5*h + 9066, 2*p + 4*h = -18 for p.
-5
Solve 3*s + 4*v + 19 = 3*v, -56*v - 1464 = 37*v + 458 - 155 for s.
0
Solve 14*h + 90 = 4*f + 24, 0 = 6*h - 3*h + f + 16 for h.
-5
Solve 4*b = -8, -3*p - 121835 = 8*b - 121873 for p.
18
Solve 0*h - 3*h - 4*m + 10 = -10*m - 17, -5*m = -9*h + 68 for h.
7
Solve 4*t - 1 = -3*s - 2, 6*s - 11 = 5*s + 10*t for s.
1
Solve -5*k = 28*w + 158, -57*w + 52*w + k = 2*k + 28 for w.
-6
Solve -2*u - 10 = -2*o, -3*o - 22*o = -u + 2*u - 27*o + 9 for u.
-1
Solve -f - 4*f + 27*r = 25*r + 13 + 24, 0 = 3*f - 5*f + 13*r - 27 for f.
-7
Solve 4*m + 55 = -67*a - 134, 3*m = -36*a + 31*a - 6 for a.
-3
Solve -4*f + 5581*w - 40 = 5577*w, -47*w + 60 = -14*f - 49*w for f.
-5
Solve 2*z + 1045 = -61*a, 3*z - 40*a = -41*a - 29 for z.
-4
Solve 0 = -5*h + 5, 10677*h - 13 = -2*i + 10676*h for i.
6
Solve -k + 19*a - 8 - 28 = 0, -4 - 4 = -5*k + a for k.
2
Solve 11 = -5*j - 80*l + 118*l, 3*j - 2*l = -69 for j.
-25
Solve -4*k = 3*f + 11, 88404*f - 88402*f - 5*k - 8 = 0 for f.
-1
Solve -l + 3 = -3*h, -10*l - 71184*h = -71186*h - 2 for l.
0
Solve -5*r + 5*f - 137 = -f, 5452*f = 4*r + 5455*f + 133 for r.
-31
Solve 3 = -4*l - 19*c + 20*c - 5, 0 = 2*l - 15*c + 62 for l.
-1
Solve -15208*l = -15211*l - 4*f - 20, 5*f + 30 = -5*l for l.
-4
Solve -5*t - 23*x = -54*x + 27*x - 188, 4*x = -2*t - 188 for t.
0
Solve 20 = -8*u - 4*s, 94*u = 89*u - 4*s - 5 for u.
-5
Solve 8*y - 84 = 13*y + 9*b - 3 + 48 - 28, 2*y + 35 = -9*b for y.
-22
Solve 3*p + 0*p - 192*q + 386*q - 50 = 195*q, 0 = 3*p - 5*q - 46 for p.
17
Solve -15*r - g = -5*g - 73, 0*r - 3*r + 12*g + 2 = 13*g for r.
3
Solve 336*w = 337*w - j - 68, -5*w - j = 38 for w.
5
Solve 5*k - 12 = 3, 30*s - 1977 = -0*s + 2*s + k - 720 for s.
45
Solve 17*g - 13*g + 447 = 83*f, -3*f + 15 = 0 for g.
-8
Solve n - h - 211 + 243 = 0, -48*n - 1306 = -2*h for n.
-27
Solve -3*z = -4*z - 3*u + 33, -z - 44 = -211729*u + 211725*u for z.
0
Solve 4120 = 60*s + 5*t, 72*t - 31*t + 12 = 38*t for s.
69
Solve -6962 - 1622 = -65*c + 2*t, 3*c + 2*t = 4*t + 411 - 11 for c.
132
Solve -5*w = 5*j - 50, 52376*j + 2210 = 52597*j + w for j.
10
Solve i + 54 = 13*n + 267, -2*i - 14*n + 74 = -18*n for i.
5
Solve 2*w - 9 = 5*q, -w - 17*q - 45 = 4*q - 7*q for w.
-3
Solve 3*y - 6*l - 243 = 0, 6142*y - 6147*y - 197 = 4*l for y.
-5
Solve -68 = -2*m + y - 26 - 36, -4*m + 5*y = - |
Expression of four growth factors in recessed extraocular muscles of rabbits.
The study was designed to determine the temporal expression of insulin-like growth factor (IGF)-I, IGF-II, basic fibroblast factor 2 (bFGF-2), and transforming growth factor beta 1 (TGF-beta1) in recessed extraocular muscles. Sixteen eyes of eight rabbits were subjected to conventional 4-mm recession of superior rectus muscles. Two rabbits were untreated as control. The rabbits were killed and their eyes were enucleated at 3 (group 3), 6 (group 6), 24 (group 24), and 72 (group 72) hours after the operation (two rabbits per group), and the expression of IGF-I, IGF-II, bFGF-2, and TGF-beta1 was immunohistochemically examined. The peak levels of IGF-I, IGF-II, and TGF-beta1 expression were observed in groups 24, 6, and 3, respectively. However, bFGF-2 was less expressed than the other growth factors in all groups. IGF-I, IGF-II, bFGF-2, and TGF-beta1 in regenerating muscle cells were expressed by different kinetics, suggesting a distinct role of each growth factor during wound healing after recession of extraocular muscles. |
<html>
<body>
<h1>Directory listing</h1>
<hr/>
<pre>
<a href="management-core-3.0.4-javadoc.jar">management-core-3.0.4-javadoc.jar</a>
<a href="management-core-3.0.4-javadoc.jar.md5">management-core-3.0.4-javadoc.jar.md5</a>
<a href="management-core-3.0.4-javadoc.jar.sha1">management-core-3.0.4-javadoc.jar.sha1</a>
<a href="management-core-3.0.4-sources.jar">management-core-3.0.4-sources.jar</a>
<a href="management-core-3.0.4-sources.jar.md5">management-core-3.0.4-sources.jar.md5</a>
<a href="management-core-3.0.4-sources.jar.sha1">management-core-3.0.4-sources.jar.sha1</a>
<a href="management-core-3.0.4.jar">management-core-3.0.4.jar</a>
<a href="management-core-3.0.4.jar.md5">management-core-3.0.4.jar.md5</a>
<a href="management-core-3.0.4.jar.sha1">management-core-3.0.4.jar.sha1</a>
<a href="management-core-3.0.4.pom">management-core-3.0.4.pom</a>
<a href="management-core-3.0.4.pom.md5">management-core-3.0.4.pom.md5</a>
<a href="management-core-3.0.4.pom.sha1">management-core-3.0.4.pom.sha1</a>
</pre>
</body>
</html>
|
Susy and Geno, Inseparable!
Susy and Geno’s long-awaited reunion finally took place on March 11 at Market-Market Mall in Taguig!
A few weeks ago, Susy started a massive search for her missing friend Geno . Susy even put up a Facebook page where all info, photos and videos in relation to the search was posted.
Finally after weeks of anticipation, Susy and Geno reunited again where the two met up not only with each other but with their loyal and very enthusiastic supporters, waving banners and placards expressing their unwavering support.
Geno arrived at the activity center holding a fresh bouquet for Susy. It was a wonderful day for Susy and Geno and for their solid fans club. After long years of waiting, the two best friends shared a long and warm embrace.
Check out this YouTube video dance performance from Susy and Geno!
The two gladly gave a dance number people requested for. Afterwards, the pair mingled with the crowd where the latter grab the chance to take photos with them.
The reunion was also the first public appearance in many years for the faces of Sustagen Milk in the 80’s and 90’s, who disappeared from the public eye, only to re-emerge two decades later, starting with Susy’s return last February. Only then would we find out that she and Geno had actually lost touch through the years.
Meanwhile, Susy and Geno’s friends from Sustagen also did their part, providing free milk for all guests and fans.
It was a lovely day for Susy and Geno and for their loyal supporters. I’m sure happy memories came to you as you watched them reunited. |
Mark,
I discussed this issue Friday with Paul, however, since it is an issue I am
not entirely knowledgeable about, I think Paul should run this by you...
---------------------- Forwarded by Tana Jones/HOU/ECT on 03/20/2000 04:44 PM
---------------------------
Paul Radous@ENRON
03/20/2000 03:12 PM
To: Tana Jones/HOU/ECT@ECT
cc:
Subject: Commodities Exchanges
Tana,
As a follow up to Friday's discoveries, on the commodities side, it appears
as though the agencies which regulate the commodity exchanges drive the rules
regarding safeguarding of client accounts. Other than 1) the US exchanges
(which are all governed by the CFTC), and 2) the OMLX, and the IPE (which we
have already addressed), what are the other exchanges whose contracts we
trade?
Thanks
Paul |
Steroid hormone modulation of olfactory processing in the context of socio-sexual behaviors in rodents and humans.
Primer pheromones and other chemosensory cues are important factors governing social interactions and reproductive physiology in many species of mammals. Responses to these chemosignals can vary substantially within and between individuals. This variability can stem, at least in part, from the modulating effects steroid and non-steroid hormones exert on olfactory processing. Such modulation frequently augments or facilitates the effects that prevailing social and environmental conditions have on the reproductive axis. The mechanisms underlying the hormonal regulation of responses to chemosensory cues are diverse. They are in part behavioral, achieved through the modulation of chemoinvestigative behaviors, and in part a product of the modulation of the intrinsic responsiveness of the main and accessory olfactory systems to conspecific, as well as other classes, of chemosignals. The behavioral and non-behavioral effects complement one another to ensure that mating and other reproductive processes are confined to reproductively favorable conditions. |
Q:
Identify slow solr queries
There are some queries that run very slow on my setup. Is there an easy way to identify and collect them (maybe through logs, or the admin console), so that I can do some performance analysis later on?
A:
yes, very easy in the logs, look at a sample line
INFO: [core0] webapp=/solr path=/select/ params={indent=on&start=0&q=*:*&version=2.2&rows=10} hits=1074 status=0 QTime=1
You need to look at Qtime
|
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VIOLENT/NON-CONSENSUAL SEX WARNING/DISCLAIMER: It is a story portraying a Conqueror/slave relationship, so it would appear non-consensual at first. As for sexual violence, there are scenes (In parts 3 and 4) which are detailed and graphic, and may not suite some readers.
Lord Conqueror of the Realm
Written by WarriorJudge
Part 19
In northern Greece , in the tavern on the border between Philippi and Macedonia , Nobleman Verosus and Nobleman Marton met with Domitia, in a room they had rented. The two Noblemen could not afford being overheard or even being seen in public with the lass.
"I don't understand. What did you do wrong?" the frustrated Nobleman Marton shouted at poor Domitia, who of no fault of her own found herself in this impossible and dangerous position. It was all Nobleman Marton could do not to resort to physical violence.
"I did exactly as I'd been told…" the young woman tried to defend herself.
Nobleman Verosus sent his fist through the wall. "Then the Conqueror should have been all over you… in and out of you!" he yelled and his eyebrows nearly touched together.
"The Conqueror wouldn't touch me," said Domitia.
Both Noblemen were still waiting for a reasonable explanation for this brilliant failure.
"Perhaps the Conqueror loves the Queen," she suggested quietly and shrugged.
Both men burst into laughter.
"Young women… All soft in the head… some of them never learn…" said Nobleman Verosus .
"Silly child," said Nobleman Marton, "the Conqueror doesn't love. The Conqueror lusts, lusts after power, lusts after blood and lusts after women, that is all. That is the source of her power. That's what sets her ever so highly above the rest of her sex. She feels no emotions and so she isn't governed by them."
"Well, the Lord Conqueror did marry the Queen," argued Domitia.
"She only married her concubine to spite us, to show us who truly rules the Empire. It is common knowledge even amongst complete idiots!"
Nobleman Marton turned to Nobleman Verosus and said, "We must consider the possibility that the Conqueror didn't take this silly girl over here because she realized it was all a ploy."
"By the Gods… what shall we do? Should we run?" Terror began to tighten its grip over Nobleman Verosus and he began fidgeting like a skittish horse.
"We are governors, we can't just disappear. Besides, there is no escaping the Conqueror. There is no place to hide, no place out of the Conqueror's reach. If we run now, the Conqueror will know we're guilty. Let me think…" Nobleman Marton said.
After some time had elapsed in silence with both men pacing restlessly from wall to wall, Nobleman Marton continued: "Lady Messalina won't say anything. She's neck deep in this and she has too much to lose."
"The Lord Conqueror knows nothing more than my name, and I am hardly the only Domitia in the Realm," she said. "And I wore nothing that would imply my station."
"That's very good. We might just come out of it alive," he said.
***
Two days had gone by. The Conqueror and the Queen were taking a stroll in the Imperial gardens, near the lily pond that the Queen adored so much. As they walked together side by side, enjoying the morning sun, the odor of blossoms and the light exercise, Gabrielle recalled the days when she had been a slave. How she used to walk in these magnificent gardens, trying to understand her Lord's moods and actions. It felt like a lifetime ago. As if to remind herself that she was in a different place now, that those days were over, Gabrielle reached for her Lord and interlaced her arm with the Conqueror's.
"They are all waiting for us in the Great Hall," Gabrielle said.
"Let them wait," the Conqueror smiled and looked at her Queen, while pressing a gentle hand over the pregnant Queen's back for support.
"There is one thing that isn't clear to me, why didn't Lady Messalina wait until after nightfall to tell me about the girl?"
"Whoever set this entire subterfuge didn't take two things into account. I wasn't familiar with the informant that disclosed Perous' whereabouts. I wasn't sure whether I could trust him or not, and I wasn't about to march blindly into a trap on the 'say so' of an informant I knew nothing about. First, I sent a scout to check the area and to confirm that Perous was indeed there and that he was alone. That took time," explained the Conqueror.
"And the second thing?"
"That I would return from Cyra alone and leave my forces behind… My desire to see you was too great. I couldn't wait."
The Queen rose to stand on her toes and placed a warm heartfelt kiss on the Conqueror's jaw, the highest place she could reach.
"You know, my Lady, you are the Realm's Sovereign."
"I know, my Lord," the Queen said and wondered why her Lord chose this time to remind her of that fact.
"And Lady Messalina is one of your ladies in waiting. She is your responsibility," the Conqueror said.
The reason for the Conqueror's words began to become apparent and clear to her. "I assume treason is punishable by death, my Lord?"
"It is, my Lady."
As they were nearing the gates of the palace, the Queen turned to the Conqueror, "My Lord?"
"Hmmm…?"
"Death is the most sever penalty for treason, is it not?" the Queen asked.
The Conqueror smiled for she understood the meaning and the reason for the Queen's question.
"It is, my Lady."
***
"The Lord Conqueror and her Majesty the Queen," the ceremony master announced as the Conqueror and the Queen entered the Great Hall.
As the Conqueror and the Queen made their way to their thrones, all present in the Great Hall bowed before them until they reached their destination and seated themselves.
"Noblemen and Ladies of the Realm," the Conqueror exclaimed, "We have summoned you all here due to a grave matter which has come to our attention and requires further investigation."
The noblemen and the ladies of the Realm began to look at one another agitatedly to see if anyone had any idea as to what the Conqueror was referring to.
"Lady Messalina," the Queen called.
Lady Messalina approached the thrones. "Your Majesties," she said and bowed before them.
As she stood before them, the Conqueror leaned over and whispered something in the Queen's ear.
"Lady Messalina, is it not true that just before noon on the day of my Lord's return from Cyra, you informed me that a young lass had been seen entering the Imperial tent?"
Lady Messalina's blood drained from her face and she grew as pale as a sheet. "It is true, your Majesty," she admitted.
"And how did you come by this bit of information?" the Queen inquired further.
"I… I can't remember, your Majesty," replied the nervous lady.
"Is it not true, that the lass in question is your very own daughter?"
Lady Messalina nearly fainted. The crowd around her gasped in surprise and walked backwards away from her, as if trying to disassociate themselves from her.
"It is, your Majesty." At this stage, lady Messalina had already realized there was no point in lying.
"Was it not your intention to cause dispute between my Lord and myself?"
Lady Messalina threw herself at the Queen's feet and began kissing them.
"You will stand up," the Queen ordered and her assertiveness gave pause to her subjects.
Lady Messalina rose back to her feet.
"You will answer the question."
"I will your Majesty," Lady Messalina replied.
"Did you act on your own volition?"
"No, your Majesty."
"Who put you up to this?" asked the Queen.
"Please, your gracious Majesty, I beg you please don't make me…"
"Nobleman Verosus and Nobleman Marton!" the Conqueror exclaimed.
Both men made their way through the crowd, mortified, joined their accomplice and bowed before the thrones.
"What have you got to say for yourselves?" the Conqueror's voice was ominous.
"Indeed not, but when her Majesty the Queen asked the question, Lady Messalina threw a glance at the two of you," said the Conqueror. "That confirmed my suspicions."
Noblemen Marton and Verosus confessed to the specifics of their scheme for all to hear by orders of the Conqueror, without trying to cast responsibility at one another and minimizing their own involvement in the traitorous conspiracy.
"Is my Lady prepared to render her verdict in the matter of Lady Messalina?" the Conqueror asked.
"I am, my Lord," the Queen replied.
"Lady Messalina, you have handled yourself poorly and reprehensibly. Being a Queen's lady in waiting is a sacred duty. It has been proven to my satisfaction that you have betrayed that duty and my trust. You have been disloyal to me and disloyal to my Lord and to the Realm. You've tried by despicable means to come between my Lord and myself. This offense I cannot and will not pardon. However, I am satisfied that there are mitigating circumstances since you were extorted. Desperation deprives some of rational thought and behooves them to take desperate measures. Therefore, it is my verdict that you should be stripped of your station and be banished from the Realm forthwith for my Lord's pleasure." The Queen voice was steady, firm and confident.
"Noblemen Marton and Verosus, greed and malice are no defense against treason. Your actions solicited, financed and facilitated an act of rebellion against us and against this Realm, which resulted in the death of several subjects and warriors of the Realm. Moreover, you have extorted her Majesty the Queen's lady in waiting and exploited her innocent daughter. You and your families will be stripped of your station and possessions. Marton and Verosus, you shall suffer a quick death in three days time. As for Macedonia , I hereby appoint Lila of Potidaea as the new governor to Macedonia and a Lady of this Realm. As for Philippi, I hereby appoint her Majesty the Queen's lady in waiting, Satrina, as the new governor to Philippi, if it pleases you, your Majesty," the Conqueror asked the Queen.
"It does, my gracious Lord," smiled the Queen.
As the guards came to remove the condemned men from the Great Hall, Lady Satrina scurried to bow before the Conqueror and the Queen.
"Your Majesties, I cannot thank you enough for your infinite kindness, honor and generosity your Majesties have shown me, and I am grateful with all my heart and soul for the great trust you place in me, but I pray you, if I may," she said and her excitement was evident in her voice.
"You may," granted the Queen.
"With your Majesties' permission, and if it pleases you, I wish to remain in her Majesty the Queen's presence and service for I am so very contented and happy with my life here in the palace," she said. "I could not have hoped to serve a kinder, nobler Sovereign than our benevolent Queen."
The Queen glanced over at the Conqueror with questioning eyes and the Conqueror, who was the one who first granted the honor, nodded her consent. Their subjects could not help but notice the silent exchange between them.
"As you wish, Lady Satrina and thank you," the Queen said and did her best to remain formal and regal and not let her own excitement be known in the forum.
"Captain Cornelius of the Imperial Guard," announced the ceremony master.
The Queen wasn't familiar with the name.
With wide determined strides, fitting a military man, Captain Cornelius approached the thrones and bowed before his Rulers.
"Your Majesties," he greeted.
It was then that the Queen recognized whom he was and fought an urge to move uncomfortably on her throne.
"With your permission, your Majesty," he humbly said and turned his attention to the Queen.
"Granted," said the Queen.
"I come before your gracious Majesty, a humble servant, to beg for forgiveness. In the past your Majesty showed me great kindness and granted excellent remedy, which I, I am ashamed to say, repaid with gross disrespect."
He chose this grand forum to offer his genuine remorse, rather than offer his apologies in private. In his mind, since he disrespected the Queen in the presence of the healer and others in the infirmary, it was only just that he should surrender his pride to the Queen in public.
He was also careful not to divulge any specifics of his transgression, including the fact that he was referring to the times back when the Queen had been a slave, so as not to cause the Queen either discomfort or embarrassment.
"I am sorry to say, I was foolish and a proud brute and I know in my heart I am not worthy of your Majesty's pardon. I assure your Majesty that as a result of your Majesty's dignity, generosity and supreme conduct towards me, which I didn't deserve I have mended my ways. I submit myself before you, your Majesty to punish as your Majesty deems fit," he said and knelt before the Queen.
"Stand up, Captain," she ordered and he obeyed.
"Your past misdeeds towards me are pardoned," the Queen said, then covered her mouth and whispered a private question in her Lord's ear, to which the latter nodded her agreement.
"You have exhibited candor and great honor, which leads me to believe your repentance is true and sincere. I hereby appoint you a nobleman to the Realm and a governor to Philippi ," the Queen said.
He lowered his head in humility and thanked his Queen for the bounty she had bestowed upon him.
"That concludes our business here today, Nobleman and Ladies of the Realm," the Conqueror stated, stood up and offered her arm to assist her pregnant Queen to her feet.
Standing in front of their subjects, the Conqueror went on to say, "As I trust you all know, today I have shown great leniency towards Marton and Verosus for their appalling treachery. By no means must you perceive it as any form of a precedent. I shall see no further division in this Realm."
As the Queen and her Lord made their way out of the Great Hall, their subjects bowed before them then began clapping their hands and chanting, " Hail to the Queen. "
Whilst strolling along the corridor that led to the Imperial chambers, the curious Queen asked, "How did my Lord know that the lass in Cyra was Messalina's daughter?"
"They have the same shade of hair color and the shape of their eyes and chins are exactly alike," the Conqueror explicated.
Alone in the privacy of their chambers, the Conqueror turned to her Queen took her small hands in hers and said with bright eyes, "I am so very proud of you, my Lady," and adorned the thin fingers with tender kisses.
***
After three days had passed, Marton and Verosus were brought to the gallows upon a wagon, which resembled one that was fit to carry small livestock. In the square stood a large crowd, as with any execution. The Conqueror always believed that even regular people, non-warriors were fascinated by death and were curious to see life as it was leaving the body. If someone else did the actual killing, then all the better.
Heavily guarded, the two men were escorted up the podium to face their Ruler and executioner. Verosus's neck was first to be stretched out and presented before the Conqueror.
As he was waiting, trembling on his knees and mumbling unintelligible words, the Conqueror unsheathed her sword, which was resting over her chiseled thigh in a leather scabbard. The polished, long and well-whetted blade caught the sun's rays.
The crowed cursed at the condemned men and cheered for their Sovereign, goading her on. It wasn't a novelty. The Conqueror knew that once she would lay the deadly strike, the cheers and the cursing would halt.
With one strike, the Conqueror put an end to his mumbling, and his severed head rolled over the floor of the podium, which was covered with sawdust to absorb the spilt blood, and his headless corpse slumped to the ground next to it.
Then came Marton's turn. Before he was shoved down to his knees by the guards and into his accomplice's pool of freshly spilt blood, the Conqueror leaned slightly towards him and whispered into his ear: “You do realize this is not retribution for some silly, inconsequential rebellion, which could have been handled quickly by a single battalion of my forces. This is mainly for trying to come between me and my Queen.”
His shocked expression was still frozen on his face when the Conqueror removed his head from his shoulders.
As the Conqueror wiped the blood off her sword and looked at Marton's head next to her boots, her mind strayed back to another execution which she had performed, the one of the British Captain, who had raped and killed some body slave, whose name the Conqueror couldn't even remember now.
Before she had sent him to his death, the Conqueror had desired to make it perfectly clear to the Captain the true and exact reason for his chastisement. When he had extended his head forward before her, whilst on his knees, she'd hissed at him, “This is for putting your filthy hands on what's mine. The slave you've raped and killed was just an excuse.” |
I welcome comments and constructive criticism of my images so that I may improve my photography
Please click on image to enlarge.
Friday, 7 October 2011
Caterpillar and Fungi.
I D's required for this caterpillar and fungi please.The caterpillar was found on the backgarden path,so no idea what plant it came from.The fungi was found under a tall bank next to a stream in the Trough of Bowland.
Christian.Thanks for your comments,I put the caterpillar on the stick and held it up with one hand and took the photo with the other. Cliff thanks for your comments. The I D is spot on thank you very much. |
Poly(ADP-ribose) polymerase (PARP) 1, whose primary role is initiation of DNA repair, is activated by damaged DNA and uses NAD+ to automodify itself and recruit other proteins involved in DNA repair. Due to its role in DNA repair, PARP-1 inhibition has been long targeted for treatment of different cancer types. By now there are already several different clinical APRP-1 inhibitors used in treatment of ovarian and breast cancers, and many others are under clinical trials for other types of cancer, such as prostate cancer, pancreatic cancer, blood cancer and others. PARP-1 inhibition has also been demonstrated to have promising effect for treatment of some cardiovascular conditions. Extensive DNA damage caused by number of cardiovascular conditions, such as a stroke or heart attack, can result in PARP-1's hyper-activation, leading to depletion of cellular NAD+ and subsequent cell death. It has been demonstrated that inhibition of PARP-1's activity using small molecules can prevent apoptosis and necrosis in such cells. Studies in animal models have indeed shown that inhibition of PARP-1 can have beneficiary effects for treatment of various cardiovascular conditions, such as ischemic stroke, cerebral ischemia, diabetic cardiomyopathy and others. Despite growing number of PARP-1 inhibitors, their molecular mechanism of action is not well understood. The overall objective of my project is to define the molecular mechanisms of activation and silencing of PARP-1. My central hypothesis is that the structural and dynamic changes occurring in PARP-1 upon DNA binding play key roles in the regulation of protein activation and dictate relative efficiency of PARP-1 inhibitors. Three specific aims are pursued in this project: 1. To define how PARP-1 is silenced through auto-modification and released from single-strand break (SSB) DNA, 2. To measure the effect of inhibitors on PARP1 structural dynamics for those that trap it at a SSB versus those that don't, 3. To define the organization and dynamics of the PARP- 1/nucleosome complex in conjunction with the housekeeping role of PARP-1 in transcriptional regulation. My proposed experiments will reveal key insights on the precise molecular mechanisms of PARP-1 activation and inhibition, aiding in the design of new PARP-1 inhibitors to improve outcomes in patients with various diseases. |
Q:
Assign values to dynamic number of sub-classes before serializing to JSON
I am integrating with a courier that requires me to pass box dimensions for each box in my consignment to their API in JSON format. I am able to set individual properties like RecipientName, but am not sure how to pass the box details for the varying number of boxes for each consignment.
The JSON needs to look like this (example is for a 2 box consignment):
{
"RecipientName": "Joe Bloggs",
"Packages" : [{
"boxNumber": "1",
"boxHeight": 1.55,
"boxLength": 1.55,
"boxWidth": 1.55
},
{
"boxNumber": "2",
"boxHeight": 2.55,
"boxLength": 2.55,
"boxWidth": 2.55
}]
}
I have built 2 classes, one that describes the structure of the JSON, and another that contains the method to serialize the JSON.
My JSON structure class looks like this (I have used a List because I have read that arrays are a fixed length, and because the number of boxes with vary I cannot use arrays):
public class API_JSON
{
public class Rootobject
{
public string RecipientName { get; set; }
public List<Package> Packages { get; set; }
}
public class Package
{
public string boxNumber { get; set; }
public double boxHeight { get; set; }
public double boxLength { get; set; }
public double boxWidth { get; set; }
}
}
And my API methods class looks like this:
public class API_Methods
{
public string recipientName;
public List<string> boxnumber;
public List<double> boxHeight;
public List<double> boxLength;
public List<double> boxWidth;
public Boolean SubmitConsignment(out string JSONData)
{
var NewRequestObject = new API_JSON.RootObject
{
Recipient = recipientName,
Packages = new API_JSON.Package
{
foreach (string item in ContainerNumber)
{
boxNumber=???,
boxHeight=???,
boxLength???=,
boxWidth=???
}
}
}
string JSONData = JsonConvert.SerializeObject(NewRequestObject);
return true;
}
}
I am then instantiating the object, setting its public variables, then running the method list this:
API_Methods myObject = new API_Methods();
myObject.recipientName;
myObject.boxnumber.Add(1);
myObject.boxnumber.Add(2);
myObject.boxHeight.Add(1.55);
myObject.boxHeight.Add(2.55);
myObject.boxLength.Add(1.55);
myObject.boxLength.Add(2.55);
myObject.boxWidth.Add(1.55);
myObject.boxWidth.Add(2.55);
bool test = API_Methods.SubmitConsignment(out JSON);
My problem is with the foreach loop - I know the code is incomplete - but I was hoping to iterate through the lists, but even with an empty foreach loop it appears to be the wrong place to put the loop as I start getting syntax errors about an expected "}"
A:
You're actually overcomplicating this for yourself - create complete package objects, and add them to the List Packages, and then pass the rootobject to the serializer.
The error you are getting is because you are not correctly initializing / filling your Packages List. Your object is invalid, hence the serializer is throwing exceptions.
This will be a lot easier for you if you create some constructors for your objects, something like this:
public Package(number, height, length, width)
{
boxNumber = number;
boxHeight = height;
//rest of your properties here in same format
}
You can then also make your setters private in the class, if you wish.
You can then easily create your package objects:
var package1 = new Package(10, 10, 10, 10);
This should make it a lot easier to create your list of boxes to put in your rootObject.
You can add each package to the packages list (individually or within a foreach loop):
Packages.Add(package1)
Or you could even start getting more concise:
Packages.Add(new Package(10,10,10,10));
You want to separate your concerns more to help keep this clear - so I'd recommend you fully construct your rootObject, add the packages to the list in one class (your 3rd code snippet), and then serialize it another (your 2nd code snippet).
Edit:
I think you'd find it easier to refactor your code somewhat:
1) Have a public rootobject in your Json_Api class, with get; set;. Get rid of the box collections. Get rid of your foreach loop from here too.
public class API_Methods
{
public rootObject RootObject { get; set; }
public Boolean SubmitConsignment(out string JSONData)
{
string JSONData = JsonConvert.SerializeObject(NewRequestObject);
return true;
}
}
2) Set the properties of this rootobject outside this class (where you currently initialize your objects). Add the New Package()s to Packages list here too.
API_Methods myObject = new API_Methods();
myObject.RootObject.recipientName = "NAME";
myObject.RootObject.Packages.Add(new Package(10,10,10,10);
myObject.RootObject.Packages.Add(new Package(20,20,20,20);
bool test = API_Methods.SubmitConsignment(out JSON);
3) Call the API method next, it should return a serialized version of the wholerootobject, including your packages.
Just a side note, it would be more conventional to send the RootObject as a parameter to the API, and return the Json string object back.
|
Brown Man of the Muirs
In the folklore of the Anglo-Scottish border the Brown Man of the Muirs is a dwarf who serves as a guardian spirit of wild animals.
Folklore
William Henderson provides an account of the Brown Man and a pair of hunters in Folklore of the Northern Counties (1879), taken from a letter sent by the historian Robert Surtees to Sir Walter Scott:
In the year before the Great Rebellion two young men from Newcastle were sporting on the high moors above Elsdon, and at last sat down to refresh themselves in a green glen near a mountain stream. The younger lad went to drink at the brook, and raising his head again saw the "Brown man of the Muirs", a dwarf very strong and stoutly built, his dress brown like withered bracken, his head covered with
frizzled red hair, his countenance ferocious, and his eyes glowing like those of a bull. After some parley, in which the stranger reproved the hunter for trespassing on his demesnes and slaying the creatures who were his subjects, and informed him how he himself lived only on whortleberries, nuts, and apples, he invited him home. The youth was on the point of accepting the invitation and springing
across the brook, when he was arrested by the voice of his companion, who thought he had tarried long, and looking round again "the wee brown man was fled." It was thought that had the young man crossed the water the dwarf would have torn him to pieces. As it was he died within the year, in consequence, it was supposed, of his slighting the dwarf's admonition, and continuing his sport on the way home.Taylor, George and Raine, James (1852). A Memoir of Robert Surtees. Durham: George Andrews. pp. 81–2.
Walter Scott in a return letter to Surtees suggested that the Brown Man may be related to the duergar (dwarfs) of Northumberland.
Fairy tales
In folklore the Brown Man appears as a solitary fairy, but in fairy tale literature he is a member of a tribe of similar beings. They once lived all over England and Scotland, but in the wake of human progress they dwindled in number and now live in a cave in Cumberland. Known as the Brown Men of the Moors and Mountains, they have great strength that allows them to hurl small boulders. By day they mine the mountains for gold and diamonds, and by night they feast in their underground hall or dance on the moors. They kidnap human children and kill any man they catch alone in the wilderness. However, they can be made subservient by repeating the incantation, "Munko tiggle snobart tolwol dixy crambo".
See also
Brownie (folklore)
Redcap
References
Category:Dwarves (mythology)
Category:English folklore
Category:Scottish folklore |
{#sp1 .466}
{#sp2 .467}
{#sp3 .468}
|
package tk.woppo.sunday.model;
import android.database.Cursor;
import com.google.gson.Gson;
import com.google.gson.annotations.SerializedName;
import java.util.HashMap;
import tk.woppo.sunday.dao.WeatherDataHelper;
import tk.woppo.sunday.dao.WeatherTodayDataHelper;
/**
* Created by Ho on 2014/7/4.
*/
public class WeatherTodayModel extends BaseModel {
private static final HashMap<String, WeatherTodayModel> CACHE = new HashMap<String, WeatherTodayModel>();
/** 城市ID */
@SerializedName("cityid")
public String id;
/** 城市名称 */
@SerializedName("city")
public String cityName;
/** 温度 */
public String temp;
/** 天气 */
public String weather;
/** 风向 */
@SerializedName("WD")
public String wind;
/** 风力 */
@SerializedName("WS")
public String ws;
/** 湿度 */
@SerializedName("SD")
public String sd;
/** 发布时间 */
public String time;
private static void addToCache(WeatherTodayModel model) {
CACHE.put(model.id, model);
}
private static WeatherTodayModel getFromCache(String id) {
return CACHE.get(id);
}
public static WeatherTodayModel fromJson(String json) {
return new Gson().fromJson(json, WeatherTodayModel.class);
}
public static WeatherTodayModel fromCursor(Cursor cursor) {
String id = cursor.getString(cursor.getColumnIndex(WeatherDataHelper.WeatherDBInfo.ID));
WeatherTodayModel model = getFromCache(id);
if (model != null) {
return model;
}
model = new Gson().fromJson(cursor.getString(cursor.getColumnIndex(WeatherTodayDataHelper.WeatherTodayDBInfo.JSON)), WeatherTodayModel.class);
addToCache(model);
return model;
}
public static class WeatherTodayRequestData {
public WeatherTodayModel weatherinfo;
}
}
|
The news of the 2015 remastering of Air Jordan retros has resulted in a load of early photos featuring next year’s Jordans. Normally at this time we’d be stuck pondering what was to come based off early product sheets and such, but this time around we’ve got high res previews of everything for your viewing pleasure. This time around: the Air Jordan 7 “French Blue”. So far the group of Spring 2015 Air Jordans has been a newer leaning group, and this retro+ colorway sticks with that trend. See the 2015 Air Jordan 7 “French Blue” below and watch for extended previews right here on Sneaker News. |
Q:
AndroidQuery ajax doesn't call php page when page contains IFRAME
I tried the following code to access my PHP page:
String url = "http://mypage.example.com/test.php?name="+data;
aq.ajax(url, String.class, new AjaxCallback<String>() {
@Override
public void callback(String url, String html, AjaxStatus status) {
Log.w(Tags.DEBUG,String.valueOf(status.getCode()));
}
});
My PHP page writes a file if the call was made. When I use the URL inside a browser, the file is created. When I use my Android app, nothing happens, the status code is 200. What else should I set?
UPDATE:
The source of my page:
<html>
<head>
<title>MY WEBSITE</title>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"></head>
<frameset rows="92, *" frameborder=NO border=1 framespacing=0 cols=*>
<frame name="topFrame" scrolling=NO noresize src="http://example.com/ads.php?ad=user12&cat=16" frameborder=NO>
<frame name="user" src="http://example.com/user12/test.php" scrolling=auto>
</frameset>
<noframes>
<body bgcolor=#FFFFFF text=#000000>
<a href="http://example.com/user12/test.php">http://example.com/user12/test.php</a>
</body>
</noframes>
</html>
A:
It seems that the web page was the problem, it sent back its source code and there was an IFRAME in it.
|
A Blog on India
Menu
Connect The Dots
In her firstbook Stay Hungry Stay Foolish Rashmi Bansal profiled twenty five entrepreneurs who were alumni of IIM – Ahmedabad. Many had then wondered including yours truly, how important an MBA degree is to become an entrepreneur. Rashmi claims this inspired her to write Connect The Dots, story of twenty one entrepreneurs but who dont have an MBA degree. The format of the book is same as her last book. There are twenty chapters, one on each entrepreneur (Gaurav Rathore & Saurabh Vyas who co founded PoliticalEDGE are covered in one chapter) and the entire chapter is based on one single interview.
The book is divided in three sections : Jugaad, Junoon & Zubaan. Jugaadis are those who didn’t get any formal training in business but learned by observing, experimenting and applying their mind. It includes some one like Kunwer Sachdev of Su-Kam who created a Rs 500 crore company from scratch; Ganesh Ram, who started what is today India’s largest English training academy, VETA when there were no BPOs and no one knew that English coaching would be as big a market as it is now.
Junoonis as the name suggests, are passionate about something that is ahead of its time. This was my favorite section in the book. Gaurav Rathore and Saurabh Vyas envisioned a consulting and research firm exclusively for politics and founded PoliticalEDGE; Satyajit Singh, founder of Shakti Sudha not only created a new industry but also benefited thousands of farmers in rural Bihar; Chetan Maini, founder of Reva, designed a solar car and has been producing electric cars since the time when global warming was not so well known and creating electric cars seemed to make little sense.
The third section Zubaan is about creative people like Paresh Mokashi, creator of Harishchandrachi Factory, India’s official entry to Oscar last year or Krishna Reddy, whose Prince Dance Group, consisting of daily wage laborers won India’s Got Talent last year.
I had great hopes from the book as I loved Stay Hungry Stay Foolish. The first chapter on Prem Ganpathy is literally a rags to riches story of someone who came to Mumbai with no money and now owns Dosa Plaza, a fast food chain with 26 outlets in the country.The rest of the stories too are very encouraging. The book is replete with inspiring anecdotes and quotes . When I read the synopsis on the third section i.e. Zubaan, I thought it would be probably the weak link in this book as stories on creatives who had made it big in the field of art would be a misfit in this book about entrepreneurs. However, all these artists achieved commercial success by following their passion and this justifies their inclusion in this book about Entrepreneurs. Entrepreneurship after all is about following your heart.
Generally when the first book is good and successful authors fail to recreate the magic in their subsequent books and that too in the same genre, as people have high expectations. In this case Rashmi Bansal definitely exceeded my expectations. A very good book and must read for some one aspires to be an entrepreneur. |
Sprint International
Sprint International may refer to:
Sprint Corporation, telecommunications company
The International (golf), golf tournament |
I soon realised that Kathy and I had settled at the periphery of the rules and the order, separated categorically from the mystics and their task; we existed like stray animals sheltered in a monastery. |
Ancient toolmaking site discovered near Niagara Falls
Archaeologists have found arrowheads and drills, indicating that the camps were occupied for extended periods of time.
DIGGING FOR TOOLS: Students at work in 2006 excavating a feature at the site on Grand Island that was most likely a hearth. (Photo: L.M. Anselmi)
An ancient campsite where people were manufacturing tools has been discovered near the Niagara Falls.
This find, combined with other archaeological discoveries in the area over the past few decades, suggests that such campsites lined the Niagara River as far back as 4,000 years ago.
So far, the team has unearthed more than 20,000 artifacts, mostly bits of rock broken off when people were creating stone tools, on the southeastern tip of Grand Island New York, about 12 miles (20 km) upstream from Niagara Falls. The earliest artifacts at the site date back at least 4,000 years, opening a window on a time when people were living a nomadic lifestyle based on hunting, fishing and gathering plants. [In Photos: Digging Up Niagara's History]
"I would anticipate that there would have been, back in the day, these kinds of campsites all along the Niagara River on both sides and on both sides of the island," team leader Lisa Anselmi, of Buffalo State University of New York, told LiveScience.
The archaeologists found that people at the Grand Island site were making a wide variety of tools, including spear points, arrowheads and even a few stone drills. Anselmi said that the drills "would be sharp enough to go through a piece of leather... or go through shell or some bone to create a bead."
The team also found bits of yellow and red ochre at the site; in ancient times it was common, for religious reasons, for ochre to be applied on the skin of someone who was being buried. No evidence of burials has been found so far at the site.
Stretching across time
The south tip of Grand Island appears to have been occupied for an extended time.
Fragments of pottery dating between 2,900 and 1,500 years ago found by Anselmi and her colleagues suggest inhabitants experimented with ceramic production, using pots to collect nuts and plant remains.
The team also found spear points that date back around 500 years, to a period shortly before Europeans started arriving in the area. More recent artifacts included nails from houses built in the 19th century and bullets that appear to date to the 1930s or 40s.
Anselmi said that the site probably would have been used mainly between the spring and fall, when food would have been plentiful. "The island would have had the advantage of being close to the river (with) lots of freshwater fish and other kinds of resources from the river," she said. Also, "in all likelihood there would have been a very strong deer population on the island."
Crossing the Niagara River
To get to Grand Island people in antiquity would have had to cross the Niagara River. Today, the fast-flowing waterway moves at a rate of about 2-3 feet per second near the island.
Curiously, rather than making use of rock found on the island, the ancient people imported a type of Onondaga chert — a tough limestone that they would have had to carry across the river from the mainland.
Anselmi explained that they would have brought over small bits of this rock that could then be molded into tools. "It's not necessarily that they're filling a canoe up with boulders," she said.
By using Onondaga chert the people of Grand Island were continuing a toolmaking tradition that goes back to when people were first entering New York State.
For instance, at a site called Emanon Pond, located in western New York, people were using the material almost exclusively nearly 11,000 years ago.
"With the exception of a single projectile point made from glacially derived drusy quartz, all of the artifacts are manufactured using local Onondaga chert," write Peter Neal Peregrine and Melvin Ember in the North America edition of the "Encyclopedia of Prehistory," published in 2001.
The findings were presented in May at a meeting of the Toronto chapter of the Ontario Archaeological Society. |
Commonwealth Bank and the Australian Chamber Orchestra kick off the 2009 Great Romantics national tour
Sydney, 11 June 2009: The Commonwealth Bank today congratulated the Australian Chamber Orchestra (ACO) on the commencement of its Great Romantics Tour.
Commonwealth Bank Group Executive Human Resources and Group Services, Ms Barbara Chapman, said the Group was committed to supporting the Arts in Australia and helping its customers, staff and the Australian community engage with music at the highest level.
“As a partner of the ACO since 1988, we have been privileged to watch it grow into the world class orchestra that it is today,” she said.
“We are proud of our ongoing support and commitment to the ACO and excited to be the 2009 National Tour Partner for the Great Romantics.”
Ms Chapman said the Commonwealth Bank was especially proud to loan its rare Guadagnini violin – crafted in 1759 in Parma, Italy, and purchased by the Bank in 1996 – to ACO’s Principal Second Violin and leader of the ACO’s Emerging Artists Program, Helena Rathbone.
“We are delighted that on the violin’s 250th birthday, it is played by such an exquisite violinist for the enjoyment and appreciation of thousands of Australians,” she said.
Ms Chapman said the Bank’s partnership with the ACO was one of three national Arts partnerships for the Group, which included Opera Australia and Bangarra Dance Theatre.
The Australian Chamber Orchestra’s Artistic Director, Mr Richard Tognetti, said he was proud of the Orchestra’s long association with the Bank.
“When I started at the ACO in 1989, the Orchestra only had a handful of corporate supporters and we were in desperate need of committed companies who would be prepared to inject cash and help fuel some new ideas,” he said.
“My dream was to create a first-rate Australian Orchestra that could hold its own anywhere in the world. The Commonwealth Bank took a risk on my dreams and, 21 years on, we have one of the most fruitful corporate relationships I’ve ever seen.”
To find out more about the Bank’s support for the Arts, visit commbank.com.au |