Behavior of FWER in Normal Distributions

Familywise error rate (FWER) has been a cornerstone in simultaneous inference for decades, and the classical Bonferroni method has been one of the most prominent frequentist approaches for controlling FWER. In a recent article, it was shown that the FWER for Bonferroni correction, under an equicorrelated multivariate normal setup asymptotically (i.e. when the number of hypotheses goes to infinity) goes to zero for any positive correlation. However, this convergence is very slow and there is very little literature on the FWER under the equicorrelated normal setup with small and moderate dimensions. The present work addresses this problem by studying the behavior of the Bonferroni FWER under the equicorrelated and general normal setups in non-asymptotic case. We also establish upper bounds on FWER in an arbitrarily correlated normal setup.