Abstract : The false discovery proportion (FDP) is a convenient way to account for
false positives when a large number $m$ of tests are performed simultaneously.
Romano and Wolf [Ann. Statist. 35 (2007) 1378-1408] have proposed a general
principle that builds FDP controlling procedures from $k$-family-wise error
rate controlling procedures while incorporating dependencies in an appropriate
manner; see Korn et al. [J. Statist. Plann. Inference 124 (2004) 379-398];
Romano and Wolf (2007). However, the theoretical validity of the latter
is still largely unknown. This paper provides a careful study of this heuristic:
first, we extend this approach by using a notion of ``bounding device''
that allows us to cover a wide range of critical values, including those
that adapt to $m_0$, the number of true null hypotheses. Second, the theoretical
validity of the latter is investigated both nonasymptotically and asymptotically.
Third, we introduce suitable modifications of this heuristic that provide
A new methods, overcoming the existing procedures with a proven FDP control.