content_image/303327.jpg content_image/303324.jpg content_image/303325.jpg
and the fact that for each j ∈ I$_{10}$, Pr( P$_{2}$$_{j}$ ≤ x ) ≤ x for all x ∈ [0 , 1] . The equality in (A.6) follows from the fact that ∪ |R$_{1}$ ( p$_{1}$ ) | r =1 C ( j ) r is a union of disjoint events, and Pr ( ∪ |R$_{1}$ ( p$_{1}$ ) | r =1 C ( j ) r | P$_{1}$ = p$_{1}$ ) = 1 . 

It follows from (A.6) that E ( ∑ j ∈ $_{I$_{10}$}$R$_{j}$ / max( R, 1) ) ≤ q$_{2}$ . Using this fact and the bound (A.3) for the first term of (A.1), we obtain: 

<img src='content_image/303327.jpg'>

## B Proof for FDR control of the oracle Procedure 3.2 

Let us now prove that under the assumption that the p -values are independent, Pro- cedure 3.2 at levels ( q $^{′}$, 2 q $^{′}$) controls the FDR at level | I$_{00}$ | ( q $^{′}$) $^{2}$/m + ( | I$_{01}$ | /m + 1) q $^{′}$. Returning to the proof of Theorem 3.2, note that (A.1) can be rewritten as follows. 

<img src='content_image/303324.jpg'>

We will now give an upper bound for each term of the sum in (B.1). First, 

<img src='content_image/303325.jpg'>

The second inequality in (B.2) follows from the facts that for each j ∈ I$_{00}$ , P$_{1}$$_{j}$ and P$_{2}$$_{j}$ are independent, and Pr( P$_{ij}$ ≤ x ) ≤ x for all x ∈ [0 , 1] and i = 1 , 2 . The equality in (B.2) follows from the explanation of the equality in (A.3). 

