  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.
