Some nonasymptotic results on resampling in high dimension, II: Multiple tests

In the context of correlated multiple tests, we aim to nonasymptotically control the family-wise error rate (FWER) using resampling-type procedures. We observe repeated realizations of a Gaussian random vector in possibly high dimension and with an unknown covariance matrix, and consider the one- and two-sided multiple testing problem for the mean values of its coordinates. We address this problem by using the confidence regions developed in the companion paper [Ann. Statist. (2009), to appear], which lead directly to single-step procedures; these can then be improved using step-down algorithms, following an established general methodology laid down by Romano and Wolf [J. Amer. Statist. Assoc. 100 (2005) 94.108]. This gives rise to several different procedures, whose performances are compared using simulated data.