Asymptotic Behaviour of Stepwise FWER-controlling Procedures

Familywise error rate (FWER) has been one of the most prominent frequentist approaches in simultaneous inference for decades, and stepwise procedures represent the most successful and ingenious attack on FWER control. A recent article proved that the FWER for the Bonferroni method asymptotically (i.e., when the number of hypotheses goes to infinity) goes to zero under any positively equicorrelated multivariate normal distribution. However, similar results for the limiting behaviors of FWER of general stepwise procedures are nonexistent. The present work addresses this problem by studying the asymptotic behavior of the FWER of step-down and step-up procedures under equicorrelated and general normality. Specifically, we show that the FWER of any step-down procedure (e.g., Holm's method) goes to zero asymptotically for a broad class of correlated normal distributions. We also establish similar results on limiting FWER for other commonly used multiple testing procedures.