Asymptotic Equivalence of Statistical Experiments

美国国家科学基金会建议DMS 0072162摘要统计实验的渐近等价性主要研究者：M。Nussbaum，康奈尔大学数学系，纽约州伊萨卡市实验的渐近等价性概念用于比较统计模型的性质。一个实验是一个家庭的概率措施;距离定义在这些对象之间，这样的信息内容的实验相对于参数是相似的，如果这个距离是小的。这个基本的缺陷伪距离（或Δ距离）已经由L。Le Cam，概括了充分性的概念。如果实验是等价的，那么它们的Δ距离是0，这里等价的Δ距离是指一个实验是对另一个实验的数据应用一个充分的近似，基于Δ距离的思想，实验的局部渐近正态性理论（LAN理论）已经发展起来，在统计学中得到了广泛的应用。数据简化的思想是充分性概念的核心，因此可以与概率的极限定理相结合，从而导致高斯移位族对一般统计模型的近似。这些高斯位移或平移族在许多情况下允许风险界的显式表达，并且这些风险界在近似模型中在渐近意义上是有效的. LAN理论的局限性在于其局限于局部设置，其中局部化率与中心极限定理中的局部化率（在大多数情况下是经典的根-n）有关。这种设置排除了应用程序的风险boundsto类不适定的函数估计问题。当前研究的重点是用Delta-距离方法处理这类问题（包括非参数密度估计），局域网理论中固有的对局部（或有限维）的限制近年来已被Brown和Low（1996）的努力以及P.I.和合作者，也开始于1996年（密度估计和高斯白色噪声的渐近等价性）。的主要工具已近似一般似然过程的Gaussianones，使用耦合方法。本项目的重点是阐述和扩展这些结果，有两个主要方向：（i）建设性地实现等价的非参数模型，使明确的等价映射（直接转移的decisionfunctions）（ii）等价的非参数模型的依赖数据（时间序列和扩散过程模型）。

NSF proposal DMS 0072162AbstractAsymptotic Equivalence of Statistical ExperimentsPrincipal Investigator: M. Nussbaum, Department of Mathematics, CornellUniversity, Ithaca, NYThe concept of asymptotic equivalence of experiments serves to compareproperties of statistical models. An experiment is a family of probabilitymeasures; a distance is defined between these objects such that theinformational content of experiments with respect to the parameter issimilar if this distance is small. This basic deficiency pseudodistance (or Delta-distance) has been introduced by L. Le Cam, generalizing theconcept of sufficiency. If experiments are equivalent via sufficiencythen their Delta-distance is 0; here equivalence via sufficiencymeans that one experiment results from application of a sufficient statisticto the data of the other.Based on the idea of the Delta-distance, a theory of localasymptotic normality (LAN-theory) of experiments has been developed whichhas found widespread application in statistics. The idea of data reductionwhich is at the heart of the sufficiency concept could thus be combined withlimit theorems of probability, resulting in approximation of generalstatistical models by Gaussian shift families. These Gaussian shift ortranslation families allow explicit expression for risk bounds in manyinstances, and these risk bound then become valid in an asymptotic sense inthe approximated models.The limitation of the LAN-theory consists in its restriction to alocal setting, in which the rate of localization is tied to thenormalization rate in the central limit theorem (the classical root-n inmost cases). This setting precludes application of the resulting risk boundsto the class of ill-posed function estimation problems. The emphasis of thecurrent project is on the treatment of this problem class (which includesnonparametric density estimation) by methods involving the Delta-distance.The restriction to a local setting (or alternatively, a finite dimensionalsetting) which is inherent in the LAN-theory has been overcome in recentyears, by efforts of Brown and Low (1996) and by variousresults of the P.I. and collaborators, starting also in 1996 (asymptotic equivalence of density estimation and Gaussian white noise). The main tool has been approximation of general likelihood processes by Gaussianones, using coupling methodology. The present project focuses on elaborationand extension of these results, with two main directions:(i) constructive realization of equivalence in nonparametric models,enabling explicit equivalence mappings (direct transfer of decisionfunctions)(ii) equivalences in nonparametric models for dependent data (time seriesand diffusion process models).