  The probability of false discovery proportion (FDP) exceeding
$\gamma\in[0,1)$, defined as $\gamma$-FDP, has received much attention as a
measure of false discoveries in multiple testing. Although this measure has
received acceptance due to its relevance under dependency, not much progress
has been made yet advancing its theory under such dependency in a nonasymptotic
setting, which motivates our research in this article. We provide a larger
class of procedures containing the stepup analog of, and hence more powerful
than, the stepdown procedure in Lehmann and Romano [Ann. Statist. 33 (2005)
1138-1154] controlling the $\gamma$-FDP under similar positive dependence
condition assumed in that paper. We offer better alternatives of the stepdown
and stepup procedures in Romano and Shaikh [IMS Lecture Notes Monogr. Ser. 49
(2006a) 33-50, Ann. Statist. 34 (2006b) 1850-1873] using pairwise joint
distributions of the null $p$-values. We generalize the notion of $\gamma$-FDP
making it appropriate in situations where one is willing to tolerate a few
false rejections or, due to high dependency, some false rejections are
inevitable, and provide methods that control this generalized $\gamma$-FDP in
two different scenarios: (i) only the marginal $p$-values are available and
(ii) the marginal $p$-values as well as the common pairwise joint distributions
of the null $p$-values are available, and assuming both positive dependence and
arbitrary dependence conditions on the $p$-values in each scenario. Our
theoretical findings are being supported through numerical studies.
