in Lemma A.1 imply Consequently, A ∗ s,r ∈ F$_{s}$ , i.e., (A.4) holds. Recall ˜ A$_{s}$ defined in (A.2). If B$_{r}$ ∈ F$_{λ}$ , then (A.4) implies (A.3), which implies ˜ A$_{s}$ ∈ F$_{s}$ , i.e., ˜ t$_{α}$ ( λ ) is a STP wrt to G if ˜ π$_{0}$ ( λ ) is measurable wrt to F$_{λ}$ . However, this holds since ˜ π$_{0}$ ( λ ) is regular. Finally, we show ˜ FDR$_{λ}$ ( ˜ t$_{α}$ ) = α , where we have written ˜ t$_{α}$ ( λ ) as ˜ t$_{α}$ . Clearly, ˜ FDR$_{λ}$ ( ˜ t$_{α}$ ) ≤ α by the definition of ˜ t$_{α}$ . Our arguments next proceed by contrapositive reasoning, i.e., that ˜ FDR$_{λ}$ ( ˜ t$_{α}$ ) < α gives a contradiction. Obviously, there must be a unique 0 ≤ j ′ ≤ n such that ˜ t$_{α}$ ∈ I$_{j}$ ′ . Since ˜ t$_{α}$ < 1, we have α < 1, ˜ π$_{0}$ ( λ ) > 0, p$_{(}$$_{j}$ $^{′}$+1) > 0 and I ∗ = [ ˜ t$_{α}$, p$_{(}$$_{j}$ $^{′}$+1) ) ⊆ I$_{j}$ ′ . Let ρ ∗ $_{m}$( t ) = m − $^{1}$L ( t ). Then ρ ∗ $_{m}$( t ) is continuous and strictly increasing on I$_{j}$ ′ by Lemma A.1. If ˜ FDR$_{λ}$ ( ˜ t$_{α}$ ) < α and α < 1, then there must be some d ′ > 0 such that ˜ I ∗ = [ ˜ t$_{α}$, ˜ t$_{α}$ + d $^{′}$] ⊆ I ∗ and that However, (A.5) implies that there exists ˆ t$_{α}$ ∈ ˜ I ∗ such that ˆ t$_{α}$ > ˜ t$_{α}$ but This contradicts the definition of ˜ t$_{α}$. Hence ˜ FDR$_{λ}$ ( ˜ t$_{α}$ ) = α must hold, which completes the whole proof. Benjamini, Y. and Y. Hochberg (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing. J. R. Statist. Soc. Ser. B 57 (1), 289–300. Chen, X. and R. Doerge (2014). Generalized estimators formultiple testing: proportion of true nulls and false discovery rate. http:// arxiv.org/abs/ 1410. 4274 . 10 References ˜ FDR$_{λ}$ ( ˆ t$_{α}$ ) = ˜ π$_{0}$ ( λ ) ρ ∗ m ( ˆ t$_{α}$ ) < α. ˜ FDR$_{λ}$ ( t ) = ˜ π$_{0}$ ( λ ) ρ ∗ $_{m}$( t ) for all t ∈ ˜ I $^{∗}$. (A.5) A ∗ s,r = A$_{s,r}$ ⋂ (⋂ n +1 j = j ∗ $_{+1}$A$_{p}$$_{(}$$_{j}$$_{)}$$_{,r}$ ) .