Resampling-based confidence regions and multiple tests for a correlated random vector

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 MIG - Unité Mathématique Informatique et Génome

We derive non-asymptotic confidence regions for the mean of a random vector whose coordinates have an unknown dependence structure. The random vector is supposed to be either Gaussian or to have a symmetric bounded distribution, and we observe $n$ i.i.d copies of it. The confidence regions are built using a data-dependent threshold based on a weighted bootstrap procedure. We consider two approaches, the first based on a concentration approach and the second on a direct boostrapped quantile approach. The first one allows to deal with a very large class of resampling weights while our results for the second are restricted to Rademacher weights. However, the second method seems more accurate in practice. Our results are motivated by multiple testing problems, and we show on simulations that our procedures are better than the Bonferroni procedure (union bound) as soon as the observed vector has sufficiently correlated coordinates.

Mots clés: resampling; cross-validation; confidence region; multiple testing; concentration inequalities; resampled quantiles; gaussian vector; unknown correlations

Domaines: Statistiques [math.ST]

Sylvain Arlot, Gilles Blanchard, Etienne Roquain. Resampling-based confidence regions and multiple tests for a correlated random vector. Nader H. Bshouty and Claudio Gentile. Learning Theory 20th Annual Conference on Learning Theory, COLT 2007, San Diego, CA, USA; June 13-15, 2007. Proceedings, Springer Berlin / Heidelberg, pp.127-141, 2007, Lecture Notes in Computer Science - Volume 4539/2007, ⟨10.1007/978-3-540-72927-3_11⟩. ⟨hal-00125670⟩