Asymptotic sharpness of product-type inequalities for maxima of random variables with applications in multiple comparisons

Product-type inequalities for the probability of the intersection of events are valid in a variety of distributional settings and have many applications in probability and statistics. These inequalities can be viewed as an improvement of the classical Bonferroni inequality. In this paper we are concerned with the problem of quantifying the asymptotic sharpness of such inequalities when they are applied to the cdf of maximum statistics. We restrict attention to the maximum of n exchangeable random variables and the range of n lid random variables as well as the studentized range. The asymptotic sharpness for n tending to infinity of the product-type inequality and Bonferroni's inequality will be determined in a variety of distributional settings. Most of them occur in multiple testing problems such as the problem of comparing n treatments with a control or, in case of the range statistic, in connection with the classical problem of pairwise comparisons of n treatments. To obtain the desired results we utilize extreme value theory and derive asymptotic expressions for the quantiles of the distributions of maxima of exchangeable random variables and of certain range statistics. As an application we investigate the true levels of some many-one and range tests based on product-type inequality adjusted critical values and study the expected number of erroneously rejected hypotheses of simultaneous multiple test procedures. (C) 2001 Elsevier Science B.V. All rights reserved.