Asymptotics in multiple hypotheses testing under dependence: beyond normality

This paper considers the classical simultaneous inference problem of testing several means in a general correlated framework. We establish an upper bound on the family-wise error rate(FWER) of Bonferroni's procedure for equicorrelated test statistics (under general distributional assumptions). Consequently, we find that for a quite general class of distributions, Bonferroni FWER asymptotically tends to zero when the number of hypotheses approaches infinity. We extend this result to general positively correlated elliptically contoured setups. Then, we establish a general theorem which holds for the class of step-down procedures under a quite general class of elliptically contoured distributions, and a wide variety of correlation structures. The results obtained in this work generalize existing results for correlated Normal test statistics and facilitate new insights into the performances of multiple testing procedures under dependence.