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A sharper Bonferroni procedure for multiple tests of significance is derived. This procedure is an improvement of Hochberg's (1988) procedure which contrasts the individual P-values with corresponding critical points. It is shown that Hochberg's original procedure is conservative, and can be made more powerful by enlarging the rejection region so that the type-one error is exactly at the nominal level. It is also shown that the modified procedure retains all the desired properties of the original procedure.

Two general classes of multiple decision functions, where each member of the first class strongly controls the family-wise error rate (FWER), while each member of the second class strongly controls the false discovery rate (FDR), are described. These classes offer the possibility that optimal multiple decision functions with respect to a pre-specified Type II error criterion, such as the missed discovery rate (MDR), could be found which control the FWER or FDR Type I error rates. The gain in MDR of the associated FDR-controlling procedure relative to the well-known Benjamini–Hochberg procedure is demonstrated via a modest simulation study with gamma-distributed component data. Such multiple decision functions may have the potential of being utilized in multiple testing, specifically in the analysis of high-dimensional data sets.