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In many applications of multiple hypothesis testing where more than one falserejection can be tolerated, procedures controlling error rates measuring atleast $k$ false rejections, instead of at least one, for some fixed $k\ge 1$can potentially increase the ability of a procedure to detect false nullhypotheses. The $k$-FWER, a generalized version of the usual familywise errorrate (FWER), is such an error rate that has recently been introduced in theliterature and procedures controlling it have been proposed. A furthergeneralization of a result on the $k$-FWER is provided in this article. Inaddition, an alternative and less conservative notion of error rate, the$k$-FDR, is introduced in the same spirit as the $k$-FWER by generalizing theusual false discovery rate (FDR). A $k$-FWER procedure is constructed given anyset of increasing constants by utilizing the $k$th order joint nulldistributions of the $p$-values without assuming any specific form ofdependence among all the $p$-values. Procedures controlling the $k$-FDR arealso developed by using the $k$th order joint null distributions of the$p$-values, first assuming that the sets of null and nonnull $p$-values aremutually independent or they are jointly positively dependent in the sense ofbeing multivariate totally positive of order two (MTP$_2$) and then discardingthat assumption about the overall dependence among the $p$-values.

Stepup procedures controlling generalized FWER and generalized FDR