control of the false discovery proportion for independently tested null hypotheses

Consider the multiple testing problem of testing m null hypotheses H1,…,Hm, among which m0 hypotheses are truly null. Given the P-values for each hypothesis, the question of interest is how to combine the P-values to find out which hypotheses are false nulls and possibly to make a statistical inference on m0. Benjamini and Hochberg proposed a classical procedure that can control the false discovery rate (FDR). The FDR control is a little bit unsatisfactory in that it only concerns the expectation of the false discovery proportion (FDP). The control of the actual random variable FDP has recently drawn much attention. For any level 1−α, this paper proposes a procedure to construct an upper prediction bound (UPB) for the FDP for a fixed rejection region. When 1−α=50%, our procedure is very close to the classical Benjamini and Hochberg procedure. Simultaneous UPBs for all rejection regions' FDPs and the upper confidence bound for the unknown m0 are presented consequently. This new proposed procedure works for finite samples and hence avoids the slow convergence problem of the asymptotic theory.