Non-asymptotic Resampling-based Confidence Regions and Multiple Tests in High Dimension

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.