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)

1 LIENS - Laboratoire d'informatique de l'école normale supérieure
2 WILLOW - Models of visual object recognition and scene understanding
3 WIAS - Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin]
4 Fraunhofer FIRST.IDA - Department Intelligent Data Analysis [Berlin]
5 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Mots clés: resampling; resampled quantile

Mots clés: confidence regions; family-wise error; multiple testing; high dimensional data; non-asymptotic error control; cross-validation; concentration inequalities

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Sylvain Arlot, Gilles Blanchard, Etienne Roquain. Some non-asymptotic results on resampling in high dimension, I: Confidence regions, II: Multiple tests. Annals of Statistics, 2010, 38 (1), pp.51-99. ⟨10.1214/08-AOS667⟩. ⟨hal-00194145v2⟩