Multi-weighting for FDR control

In the context of multiple hypothesis testing, we propose a new method of $p$-value weighting when controlling the false discovery rate (FDR). To deal with all the possible rejection numbers in the FDR, we consider a matrix of weights and we integrate it in a new class of procedures, called the "multi-weighted" procedures. First, we propose multi-weighted step-up and step-down procedures which control rigourously the FDR under independence, positive dependence (PRDS) and general dependence. Second, we propose a way to derive a weight matrix which is optimal in the sense of power. When the optimal weight matrix is reasonably well approached, we demonstrate in a simulation study that the corresponding multi-weighted procedures outperform the standard uniformly-weighted Benjamini-Hochberg procedure. Finally, we apply our method to the two sample multiple comparison problem, and we demonstrate its interest in practice with an application to microarray experiments.