Stopping time property of thresholds of Storey-type FDR procedures

For multiple testing, we introduce Storey-type FDR procedures and the concept of "regular estimator of the proportion of true nulls". We show that the rejection threshold of a Storey-type FDR procedure is a stopping time with respect to the backward filtration generated by the p-values and that a Storey-type FDR estimator at this rejection threshold equals the pre-specified FDR level, when the estimator of the proportion of true nulls is regular. These results hold regardless of the dependence among or the types of distributions of the p-values. They directly imply that a Storey-type FDR procedure is conservative when the null p-values are independent and uniformly distributed.