  The concept of $k$-FWER has received much attention lately as an appropriate
error rate for multiple testing when one seeks to control at least $k$ false
rejections, for some fixed $k\ge 1$. A less conservative notion, the $k$-FDR,
has been introduced very recently by Sarkar [Ann. Statist. 34 (2006) 394--415],
generalizing the false discovery rate of Benjamini and Hochberg [J. Roy.
Statist. Soc. Ser. B 57 (1995) 289--300]. In this article, we bring newer
insight to the $k$-FDR considering a mixture model involving independent
$p$-values before motivating the developments of some new procedures that
control it. We prove the $k$-FDR control of the proposed methods under a
slightly weaker condition than in the mixture model. We provide numerical
evidence of the proposed methods' superior power performance over some $k$-FWER
and $k$-FDR methods. Finally, we apply our methods to a real data set.
