Procedures controlling generalized false discovery rate

Procedures controlling error rates measuring at least k false rejections, instead of at least one, can potentially increase the ability of a procedure to detect false null hypotheses in situations where one seeks to control k or more false rejections having tolerated a few of them. The k-FWER, which is the probability of at least k false rejections and generalizes the usual familywise error rate (FWER), is such an error rate that is recently introduced in the literature and procedures controlling it have been proposed. This paper considers an alternative and less conservative notion of error rate, the k-FDR, which is the expected proportion of k or more false rejections among all rejections and generalizes the usual notion of false discovery rate (FDR). Procedures with the k-FDR control dominating the Benjamini-Hochberg stepup FDR procedure and its stepdown analog under independence or positive dependence and the Benjamini-Yekutieli stepup FDR procedure under any form of dependence have been constructed.