Consistent estimation of the proportion of false nulls and FDR for adaptive multiple testing Normal means under weak dependence

We consider multiple testing means of many dependent Normal random variables that do not necessarily follow a joint Normal distribution. Under weak dependence, we show the uniform consistency of proportion estimators that are constructed as solutions to Lebesgue-Stieltjes equations for the setting of a point, bounded and one-sided null, respectively, and characterize via the index of weak dependence the sparsest proportion these estimators can consistently estimate. On the other hand, under a principal correlation structure and employing a suitable definition of p-value for composite null hypotheses, we show that three key empirical processes induced by a single-step multiple testing procedure (MTP) satisfy the strong law of large numbers for testing each of the three types of nulls. Further, under this structure and for testing a point null and a one-sided null respectively, we construct an adaptive single-step MTP that employs a proportion estimator mentioned earlier, and show that the false discovery proportion of this procedure satisfies the weak law of large numbers and hence consistently estimates the false discovery rate of the procedure. In addition, we report some findings on the estimators of Jin and of Meinshausen and Rice of the proportion of false nulls in the critically and very sparse regimes under weak dependence and model misspecifications, respectively.