HAL : hal-00194145, version 2

arXiv : 0712.0775

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The Annals of Statistics 38, 1 (2010) 51-99

Versions disponibles :

v1 (05-12-2007)

v2 (06-07-2009)

Some non-asymptotic results on resampling in high dimension, I: Confidence regions, II: Multiple tests

Sylvain Arlot 1, 2, Gilles Blanchard 3, 4, Etienne Roquain 5

(2010)

We study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure. The random vector is supposed to be either Gaussian or to have a symmetric and bounded distribution. The dimensionality of the vector can possibly be much larger than the number of observations and we focus on a non-asymptotic control of the confidence level, following ideas inspired by recent results in learning theory. We consider two approaches, the first based on a concentration principle (valid for a large class of resampling weights) and the second on a direct resampled quantile, specifically using Rademacher weights. Several intermediate results established in the approach based on concentration principles are of self-interest. We also discuss the question of accuracy when using Monte-Carlo approximations of the resampled quantities. We present an application of these results to the one-sided and two-sided multiple testing problem, in which we derive several resampling-based step-down procedures providing a non-asymptotic FWER control. We compare our different procedures in a simulation study, and we show that they can outperform Bonferroni's or Holm's procedures as soon as the observed vector has sufficiently correlated coordinates.

1 :	Laboratoire d'informatique de l'école normale supérieure (LIENS)
	CNRS : UMR8548 – Ecole Normale Supérieure de Paris - ENS Paris
2 :	WILLOW (INRIA Rocquencourt)
	INRIA – Ecole Normale Supérieure de Paris - ENS Paris – Ecole des Ponts ParisTech – CNRS : UMR8548
3 :	Weierstrass institute for applied stochastics and analysis (WIAS)
	WIAS
4 :	Fraunhofer FIRST.IDA (FHG FIRST.IDA)
	Fraunhofer Institute
5 :	Laboratoire de Probabilités et Modèles Aléatoires (LPMA)
	CNRS : UMR7599 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot

Domaine

:

Mathématiques/Statistiques Statistiques/Théorie

Mots Clés : confidence regions – family-wise error – multiple testing – high dimensional data – non-asymptotic error control – resampling – cross-validation – concentration inequalities – resampled quantile

hal-00194145, version 2

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Contributeur : Sylvain Arlot <>

Soumis le : Lundi 6 Juillet 2009, 11:25:18

Dernière modification le : Jeudi 1 Juillet 2010, 14:35:30