  In many applications of multiple hypothesis testing where more than one false
rejection can be tolerated, procedures controlling error rates measuring at
least $k$ false rejections, instead of at least one, for some fixed $k\ge 1$
can potentially increase the ability of a procedure to detect false null
hypotheses. The $k$-FWER, a generalized version of the usual familywise error
rate (FWER), is such an error rate that has recently been introduced in the
literature and procedures controlling it have been proposed. A further
generalization of a result on the $k$-FWER is provided in this article. In
addition, an alternative and less conservative notion of error rate, the
$k$-FDR, is introduced in the same spirit as the $k$-FWER by generalizing the
usual false discovery rate (FDR). A $k$-FWER procedure is constructed given any
set of increasing constants by utilizing the $k$th order joint null
distributions of the $p$-values without assuming any specific form of
dependence among all the $p$-values. Procedures controlling the $k$-FDR are
also 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 are
mutually independent or they are jointly positively dependent in the sense of
being multivariate totally positive of order two (MTP$_2$) and then discarding
that assumption about the overall dependence among the $p$-values.
