On improving some adaptive BH procedures controlling the FDR under dependence

Blanchard and Roquain (2009) presented for the first time methods of adapting the Benjamini-Hochberg (BH) method to data through an estimate of the proportion of true null hypotheses that continue to control the false discovery rate (FDR) under positive dependence in a non-asymptotic setting. However, they are often too conservative to provide a real improvement of the BH method. To obtain adaptive BH methods with proven FDR control improving the original BH method in more situations than what are seen in Blanchard and Roquain (2009), we propose alternative versions of the Blanchard-Roquain methods under some additional assumptions allowing explicit use of pairwise correlations whenever they are available. We offer numerical evidence of improved performances of the proposed alternatives in two scenarios involving test statistics satisfying the positive dependence conditions assumed for the main results.