Behaviour of Familywise Error Rate in Normal Distributions

We study the behaviour of the familywise error rate (FWER) for Bonferroni-type procedure in multiple testing problem. Das and Bhandari in a recent article have shown that, in the equicorrelated normal setup, FWER asymptotically (i.e when number of hypotheses is very large) is a convex function of correlation $\rho$ and hence an upper bound on the FWER of Bonferroni-$\alpha$ procedure is given by $\alpha(1 - \rho)$. We derive upper bounds on FWER for Bonferroni method under the equicorrelated and general normal setups in asymptotic and non-asymptotic case. We show similar results for generalized familywise error rates.