Asymptotic Equivalence of Statistical Experiments

NSF提案DMS 0072162统计试验的渐近等价性主要研究人员：M.Nussbaum，康奈尔大学数学系，伊萨卡，NY试验的渐近等价性的概念用于比较统计模型的性质。实验是一族概率度量；定义了这些对象之间的距离，使得如果该距离较小，则实验相对于参数的信息内容是相似的。这一基本亏伪距离(或Delta距离)是由L.Le Cam引入的，推广了充分性的概念。如果实验是通过充分性等价的，那么它们的增量距离是0；这里通过充分性等价是指一个实验是通过对另一个实验的数据应用充分的统计而得到的。基于增量距离的思想，发展了实验的局部渐近正态理论(局域网理论)，该理论在统计学中得到了广泛的应用。因此，作为充分性概念核心的数据约简的思想可以与概率极限定理相结合，从而产生用高斯移动族逼近一般统计模型的结果。这些高斯移动族或平移族允许在许多情况下显式地表示风险界，然后这些风险界在近似模型中变得在渐近意义上有效。局域网理论的局限性在于其对局部设置的限制，其中局部化速度与中心极限定理中的正规化速度(在大多数情况下是经典的根n)有关。这种设置排除了将所得到的风险边界应用于一类不适定函数估计问题。当前项目的重点是通过涉及Delta距离的方法来处理这类问题(包括非参数密度估计)。近年来，通过Brown和Low(1996)的努力以及P.I.和合作者的各种结果(密度估计和高斯白噪声的渐近等价性)，局域网理论中固有的对局部设置(或有限维设置)的限制已经被克服。主要的工具是用耦合方法用高斯酮逼近一般似然过程。本项目的重点是阐述和推广这些结果，主要有两个方向：(I)在非参数模型中建设性地实现等价，使得显式等价映射(决策函数的直接转移)成为可能；(Ii)相依数据(时间序列和扩散过程模型)在非参数模型中的等价。

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).