Adaptive multiple testing procedures under positive dependence

In multiple testing, the unknown proportion of true null hypotheses among all null hypotheses that are tested often plays an important role. In adaptive procedures this proportion is estimated and then used to derive more powerful multiple testing procedures. Hochberg and Benjamini (1990) first presented adaptive Holm and Hochberg procedures for controlling the familywise error rate (FWER). However, until now, no mathematical proof has been provided to demonstrate that these procedures control the FWER in finite samples. In this paper, we present new adaptive Holm and Hochberg procedures and prove they can control the FWER in finite samples under some common types of positive dependence. Through a small simulation study, we illustrate that these adaptive procedures are more powerful than the corresponding non-adaptive procedures.