holds for general p-values is a new result, whose consequences will be discussed in Section 4; (iii) the validity of the STP of ˜ t$_{α}$ ( λ ), regardless of the joint distribution of the p-values, depends crucially on the intrinsic structure of the backward filtration G ; (iv) the STP of the threshold (wrt a similar backward filtration) defined by equation (7.2) in Pena et al. (2011) does hold trivially due to the special structure of the filtration there. We give a few implications of Theorem 1. For concise statements, we set assumption A1) as “the null p-values are independent and uniformly distributed”. Further, for each t ∈ [0 , 1) let H$_{t}$ = σ ( 1$_{{}$$_{p}$$_{i}$$_{≤}$$_{s}$$_{}}$ , 0 ≤ s ≤ t, i = 1 , ..., m ) . Proof. The first claim is a direct consequence of Theorem 1. We show the second claim. By Corollary 1 in Liang and Nettleton (2012), E [ ˜ F DR$_{λ}$ ( ˜ t$_{α}$ ) ] ≥ F DR ( ˜ t$_{α}$ ) . However, Theorem 1 implies E [ ˜ F DR$_{λ}$ ( ˜ t$_{α}$ ) ] = α . So, F DR ( ˜ t$_{α}$ ) ≤ α . Finally, we show the third. By Theorem 4 in Liang and Nettleton (2012), E [ ˜ F DR$_{λ}$ ˜ t$_{α}$ ] ≥ F DR ( ˜ t$_{α}$ ) . Therefore, from Theorem 1 we see F DR ( ˜ t$_{α}$ ) ≤ α . This completes the proof. Corollary 1 illustrates the connection between the regularity of the estimator of the propor- tion of true nulls, the STP of the rejection threshold, and the conservativeness of a Storey-type FDR procedure. The third conclusion in Corollary 1 allows to choose λ in ˜ π$_{0}$ ( λ ) from the data in a Storey-type FDR procedure without potentially sacrificing its conservativeness. Further, it 5 3. If λ is a stopping time wrt to {H$_{t}$ : t ∈ [0 , 1) } and lies in a compact subset [ κ, τ ] in (0 , 1) for some κ < τ and ˜ π$_{0}$ ( λ ) is regular, then a Storey-type FDR procedure is conservative when ˜ t$_{α}$ ≤ κ . 2. Storey’s procedure in Storey et al. (2004) is conservative when ˆ π$_{0}$ ( λ ) in (4) is regular and ˜ t$_{α}$ ( λ ) < 1 . 1. Lemma 4 of Storey et al. (2004) is valid when ˆ π$_{0}$ ( λ ) in (4) is regular. Corollary 1. Assume A1), the following hold: 4 Conservativeness of Storey-type FDR procedures