On k-FWE-based critical values for controlling the false discovery proportion under dependence

The false discovery proportion (FDP) is a convenient way to account for false positives in an high dimensional setting where a large number of tests are performed simultaneously. The Benjamini-Hochberg procedure is now widely used and is known to control the expectation of the FDP, called the false discovery rate (FDR). However, when the individual tests are correlated, controlling the FDR can be unsuitable to ensure that the actually achieved FDP is close (or below) the targeted level. This rises the question of controlling the quantiles of the distribution of the FDP, which is a challenging question that has received a growing attention in the recent literature. This paper elaborates upon the general principle let down by Romano and Wolf (2007) (RW) that builds FDP controlling procedures from $k$-family-wise error rate ($k$-FWE) controlling procedures, while incorporating known dependencies in an appropriate manner. This method is revisited as follows: first, choose a device to upper-bound the $k$-FWE, for all $k$. Second, build the corresponding critical values, possibly adaptively to the number $m_0$ of true null hypotheses. Third, use these critical values into a step-wise procedure (either step-down or step-up). The goal of the paper is to study the obtained FDP when using this methodology. Our first result provides sample finite bounds, while our second result is asymptotic in the number $m$ of hypotheses. Overall, this paper can be seen as a validation of RW's paradigm for controlling the FDP under dependence.