Resampling-based confidence regions and multiple tests for a correlated random vector
Sylvain Arlot, Gilles Blanchard and Etienne Roquain
Lecture Notes in Artifical Intelligence
We study generalized bootstrapped confidence regions for the mean of a random vector whose coordinates have an unknown dependence structure, with 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 boostrapped quantile. The first one allows us 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.