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Philippe Berthet, U. Paul Sabatier Print
Thursday, 16 May 2019, 12:15 - 13:15

Philippe Berthet, Université Paul Sabatier 

Auxiliary information empirical processes and their bootstrap

Abstract : In non parametric statistics, when some information on the unknown distribution is available but not enough to justify a parametric or semi parametric model, a crucial question is : how to exploit the auxiliary information ? The empirical measure has to be modified to match the information by using projection, empirical likelyhood or plug-in. Such weight changes could be sequential to follow the online updating or fast learning of the auxiliary information, for instance in a distributed data setting. However, if the information is of mixed nature and sources, the change of empirical mesure is not obvious, in particular the appropriated new process could be no more a centered measure. We will present in details the iterative raking-ratio procedure, often used but not studied nor justified since Kullback identified the minimum contrast limit. With M. Albertus we obtain asymptotic and non asymptotic results for a fixed number of iterations - then infinite. The main fact is the mathematical justification of the uniform decrease of biais, variance, covariance and quadratic risk over large classes of linear estimators - through the empirical process indexed by functions - and the closed form expression of the limiting auxiliary information Gaussian process - that is no more a Brownian Bridge. Uniform Berry-Esseen type results and concentration probability bounds follow, thanks to the coupling between the raking-ratio empirical process and its limiting process. In order to evaluate the decrease of variances and risk, it is natural to bootstrap the raked statistics. Some precise results on the weighted bootstrap will also be presented, when the law of each random weight may depend on the sample point. By evaluating the distance in distribution between the bootstrapped samples, the initial sample and their joint limiting processes we provide the allowed size of monte-carlo bootstraps allowing a control of the unavoidable bias and variance distorsion. Even in the usual setting without auxiliary information the results of Giné and Zinn are then extended and strenghened with rates and joint approximations. To illustrate the raking-ratio approach we will consider the case of a few known quantiles in the estimation of mutivariate directional quantiles and Kantorovitch-Wasserstein type distances.

 

Location: R42.2.113
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