Optimality of the Holm procedure among general step-down multiple testing procedures

We study the class of general step-down multiple testing procedures, which contains the usually considered procedures determined by a nondecreasing sequence of thresholds (we call them threshold step-down, or TSD, procedures) as a parametric subclass. We show that all procedures in this class satisfying the natural condition of monotonicity   and controlling the family-wise error rate (FWER) at a prescribed level are dominated by one of them — the classical Holm procedure. This generalizes an earlier result pertaining to the subclass of TSD procedures [Lehmann, E.L., Romano, J.P., 2005. Testing Statistical Hypotheses, 3rd ed. Springer, New York]. We also derive a relation between the levels at which a monotone step-down procedure controls the FWER and the generalized FWER (the probability of kk or more false rejections).