Unimprovability of the Bonferroni procedure in the class of general step-up multiple testing procedures

We introduce the class of general step-up multiple testing procedures (step-up MTPs), which contains the usually considered Benjamini-Hochberg type procedures (we call them threshold step-up MTPs) as a parametric subclass. We show that, under the natural condition of monotonicity, the Bonferroni procedure cannot be improved on, while controlling the family-wise error rate (FWER) at the same level, in the class of step-up procedures. This is in contrast to the class of step-down MTPs, where the Holm procedure is a classic example of such an improvement.