Summary: We consider the problem of testing multiple null hypotheses. A classical approach to dealing with the multiplicity problem is to restrict attention to procedures that control the familywise error rate (FWER), the probability of even one false rejection. However, if \(s\) is large, control of the FWER is so stringent that the ability of a procedure which controls the FWER to detect false null hypotheses is limited. Consequently, it is desirable to consider other measures of error control. We consider methods based on control of the false discovery proportion (FDP) defined by the number of false rejections divided by the total number of rejections (defined to be \(0\) if there are no rejections). The false discovery rate proposed by Y. Benjamini and Y. Hochberg [J. R. Stat. Soc., Ser. B 57, No. 1, 289–300 (1995; Zbl 0809.62014)] controls E(FDP). We construct methods such that, for any \(\gamma\) and \(\alpha\), \(P\{FDP>\gamma\}\leq\alpha\). Based on \(p\)-values of individual tests, we consider stepdown procedures that control the FDP, without imposing dependence assumptions on the joint distribution of the \(p\)-values. A greatly improved version of a method given by E.L. Lehmann and J.P. Romano [Ann. Stat. 33, No. 3, 1138–1154 (2005; Zbl 1072.62060)] is derived and generalized to provide means by which any sequence of nondecreasing constants can be rescaled to ensure control of the FDP. We also provide a stepdown procedure that controls the FDR under a dependence assumption.
For the entire collection see [Zbl 1113.62002].