A smooth version of the step-up procedure for multiple tests of hypotheses

Cohen and Sackrowitz [2005. Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure. Ann. Statist. 33, 145-158; 2007. More on the inadmissibility of step-up. J. Multivariate Anal. 97, 481-492] have demonstrated that the popular step-up (SU) multiple testing procedure is inadmissible under a wide variety of conditions. All conditions, however, did assume a permutation invariant (symmetric) model. In this paper we find a necessary condition for admissibility of multiple testing procedures in the asymmetric case. Once again SU does not satisfy the condition and is inadmissible. Since SU has a somewhat less favorable practical property and a less favorable theoretical property, we offer a smooth version of SU which retains the favorable practical properties and avoids some of the less favorable ones. In terms of performance the smooth version and nonsmooth version seem to be comparable at least in low dimensions.