Non-asymptotic resampling-based confidence regions and multiple tests in high dimension

Sylvain Arlot (1, 2) , Gilles Blanchard (3) , Etienne Roquain (4)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 SELECT - Model selection in statistical learning
3 Fraunhofer FIRST.IDA - Department Intelligent Data Analysis [Berlin]
4 Department of Mathematics [Amsterdam]

We study generalized bootstrapped confidence regions for the mean of a random vector whose coordinates have an unknown dependence structure. 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. The random vector is supposed to be either Gaussian or to have a symmetric bounded distribution. We consider two approaches, the first based on a concentration principle and the second on a direct bootstrapped quantile. The first one allows us to deal with a very large class of resampling weights while our results for the second are specific to Rademacher weights. 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: confidence regions; family-wise error; multiple testing; high dimensional data; non-asymptotic error control; resampling; cross-validation; concentration inequalities; resampled quantile

Domaines: Statistiques [math.ST]

Sylvain Arlot, Gilles Blanchard, Etienne Roquain. Non-asymptotic resampling-based confidence regions and multiple tests in high dimension. 2007. ⟨hal-00194145v1⟩