Asymptotic Equivalence of Statistical Experiments

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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).