Stopping time property of thresholds of Storey-type FDR procedures

Controlling the false discovery rate (FDR) in multiple testing has been proven to be more powerful when there are hundreds or even millions of hypotheses to test. Various FDR procedures have been developed based on p-values, and at a desired FDR level each such procedure rejects all null hypotheses whose associated p-values are no larger than a threshold. As the critical ingredient that induces the decision rule and determines if the FDR of an FDR procedure is no larger than the targeted level, the threshold is very important and its behavior needs to be fully understood. We show that the threshold of a Storey-type FDR procedure is a stopping time with respect to the backward filtration generated by the p-values, regardless of the dependence among or the types of distributions of the p-values. We also show that each Storey-type estimator at the threshold of the corresponding Storey-type FDR procedure equals the pre-specified FDR level.