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

Sylvain Arlot; Gilles Blanchard; Etienne Roquain

hal-00194145, version 2

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

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

Abstract: 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 Analysis and Stochastics (WIAS)
Forschungsverbund Berlin e.V. (FVB)
4: Fraunhofer FIRST.IDA (FHG FIRST.IDA)
Fraunhofer Institute
5: Laboratoire de Probabilités et Modèles Aléatoires (LPMA)
CNRS : UMR7599 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot

hal-00194145, version 2
http://hal.archives-ouvertes.fr/hal-00194145
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From: Sylvain Arlot <>
Submitted on: Monday, 6 July 2009 11:25:18
Updated on: Thursday, 1 July 2010 14:35:30

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%% hal-00194145, version 2
%% http://hal.archives-ouvertes.fr/hal-00194145
@article{arlot:hal-00194145,
    hal_id = {hal-00194145},
    url = {http://hal.archives-ouvertes.fr/hal-00194145},
    title = {{Some non-asymptotic results on resampling in high dimension, I: Confidence regions, II: Multiple tests}},
    author = {Arlot, Sylvain and Blanchard, Gilles and Roquain, Etienne},
    abstract = {{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.}},
    keywords = {confidence regions ; family-wise error ; multiple testing ; high dimensional data ; non-asymptotic error control ; resampling ; cross-validation ; concentration inequalities ; resampled quantile},
    language = {English},
    affiliation = {Laboratoire d'informatique de l'{\'e}cole normale sup{\'e}rieure - LIENS , WILLOW - INRIA Rocquencourt , Weierstrass Institute for Applied Analysis and Stochastics - WIAS , Fraunhofer FIRST.IDA - FHG FIRST.IDA , Laboratoire de Probabilit{\'e}s et Mod{\`e}les Al{\'e}atoires - LPMA},
    pages = {51-99},
    journal = {The Annals of Statistics},
    volume = {38},
    number = {1 },
    note = {long version of arXiv:math/0701605 This work was supported in part by the IST and ICT Programmes of the European Community, successively under the PASCAL (IST-2002-506778) and PASCAL2 (ICT-216886) networks of excellence. },
    audience = {international },
    year = {2010},
    pdf = {http://hal.archives-ouvertes.fr/hal-00194145/PDF/ABR09\_1\_RC.pdf, http://hal.archives-ouvertes.fr/hal-00194145/PDF/ABR09\_2\_T.pdf},
}

%F hal-00194145, version 2
%0 Journal Article
%U http://hal.archives-ouvertes.fr/hal-00194145
%2 MATH:MATH_ST;STAT:TH
%T Some non-asymptotic results on resampling in high dimension, I: Confidence regions, II: Multiple tests
%A Arlot, Sylvain
%A Blanchard, Gilles
%A Roquain, Etienne
%+ Laboratoire d'informatique de l'école normale supérieure - LIENS , WILLOW - INRIA Rocquencourt , Weierstrass Institute for Applied Analysis and Stochastics - WIAS , Fraunhofer FIRST.IDA - FHG FIRST.IDA , Laboratoire de Probabilités et Modèles Aléatoires - LPMA
%X 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.
%G English
%J The Annals of Statistics
%2 international
%8 2010-00-00
%2 2010
%V 38
%N 1
%P 51-99
%K confidence regions ; family-wise error ; multiple testing ; high dimensional data ; non-asymptotic error control ; resampling ; cross-validation ; concentration inequalities ; resampled quantile
%Z AMS classification: Primary 62G15 ; secondary 62G09, 62G10
%Z long version of arXiv:math/0701605
%Z This work was supported in part by the IST and ICT Programmes of the European Community, successively under the PASCAL (IST-2002-506778) and PASCAL2 (ICT-216886) networks of excellence.

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