covers the dynamic Storey’s FDR procedure in Liang and Nettleton (2012) where ˆ π$_{0}$ ( λ ) in (4) is used and λ is determined from the data. 

Compared to the proofs of the conservativeness of (dynamic) Storey’s procedure in Storey et al. (2004) and Liang and Nettleton (2012), the proof of Corollary 1 we provide based on Theorem 1 is much shorter and the second conditioning step there is no longer needed. This illustrates the usefulness of our finding (6) for general p-values in the martingale arguments to prove the conservativeness of a Storey-type FDR procedure. 

## 5 Discussion 

We have shown the STP of the rejection threshold of a Storey-type FDR procedure for gen- eral p-values, that a Storey-type FDR estimator equals the targeted FDR level at its rejection threshold, and that a Storey-type FDR procedure is usually conservative, when the estimator of the proportion of true nulls is regular. This implies that the STP may not hold for the rejection threshold of an adaptive FDR procedure when an estimator of the proportion of true nulls is not adaptive to the backward filtration. Therefore, caution should be taken before applying martingale arguments to prove the conservativeness of an adaptive FDR procedure. In view of the increasing use of stochastic processes (such as decision processes) and suprema related to these processes (such as rejection thresholds) in multiple testing, to better understand the behavior of these suprema remains an important task. 

## Acknowledgements 

This research was funded by a National Science Foundation Plant Genome Research Program grant (No. IOS-1025976) to RWD. Its major part was completed when X. Chen was a PhD student at the Department of Statistics, Purdue University. 

