Inequalities between generalized familywise error rates of a multiple testing procedure

We consider a multiple testing procedure (MTP) that decides which hypotheses to reject based solely on the observed p-values associated with the hypotheses being tested. Let fk be the exact level at which the MTP weakly controls, under a general and unknown dependence structure of the p-values, the kth generalized familywise error rate--the probability of k or more false rejections. The sequence f1,...,fm, where m is the number of hypotheses being tested, is nonincreasing. We show that if the MTP is monotone (reducing p-values can only increase the number of rejections), then the sequence kfk is nondecreasing. This result pertaining to the weak control of generalized FWERs (all hypotheses are true) carries over to the situation where a limited number of hypotheses may be false.