Adaptive FDR control under independence and dependence

In the context of multiple hypotheses testing, the proportion $π_0$ of true null hypotheses in the pool of hypotheses to test often plays a crucial role, although it is generally unknown a priori. A testing procedure using an implicit or explicit estimate of this quantity in order to improve its efficency is called adaptive. In this paper, we focus on the issue of False Discovery Rate (FDR) control and we present new adaptive multiple testing procedures with control of the FDR. First, in the context of assuming independent $p$-values, we present two new procedures and give a unified review of other existing adaptive procedures that have provably controlled FDR. We report extensive simulation results comparing these procedures and testing their robustness when the independence assumption is violated. The new proposed procedures appear competitive with existing ones. The overall best, though, is reported to be Storey's estimator, but for a parameter setting that does not appear to have been considered before. Second, we propose adaptive versions of step-up procedures that have provably controlled FDR under positive dependences and unspecified dependences of the $p$-values, respectively. While simulations only show an improvement over non-adaptive procedures in limited situations, these are to our knowledge among the first theoretically founded adaptive multiple testing procedures that control the FDR when the $p$-values are not independent.