A Proof of Theorem 3.2 Note that given P$_{1}$ , for r > |R 1 | , C ( j ) r =  , since exactly |R 1 | − 1 T$_{i}$ ’s are finite. Obviously, C ( j ) r and C ( j ) r ′ are disjoint events for any r = r $^{′}$, and ∪ m r $_{=1}$C ( j ) r is the entire space of ( P ( j ) 1 , P ( j ) 2 ). Let I$_{0}$ = I$_{01}$ ∪ I$_{00}$ , R$_{j}$ be the indicator of whether H$_{NR,j}$ was rejected for j = 1 , . . . , m , and R = ∑ m j $_{=1}$R$_{j}$ . The FDR for the family of no replicability null hypotheses is Let T$_{(1)}$ ≤ . . . ≤ T$_{(}$$_{m}$$_{−}$$_{1)}$ be the sorted T -values, and T$_{(0)}$ = 0. For r = 1 , . . . , m , we define C ( j ) r as the event in which if H$_{NR,j}$ is rejected by Procedure 3.2, r hypotheses are rejected including H$_{NR,j}$ : T$_{i}$ =     max ( mP$_{1}$$_{i}$ q$_{1}$ , ( |R ( j ) 1 ( P ( j ) 1 ) | +1) P$_{2}$$_{i}$ q$_{2}$ ) if i ∈ R ( j ) 1 ( P ( j ) 1 ) , ∞ otherwise. C ( j ) r = { ( P ( j ) 1 , P ( j ) 2 ) : T$_{(}$$_{r}$$_{−}$$_{1)}$ ≤ r, T$_{(}$$_{r}$$_{)}$ > r + 1 , T$_{(}$$_{r}$$_{+1)}$ > r + 2 , . . . , T$_{(}$$_{m}$$_{−}$$_{1)}$ > m } . F DR = E ( ∑ j ∈ I$_{0}$ R$_{j}$ max( R, 1) ) + E ( ∑ j ∈ I$_{10}$ R$_{j}$ max( R, 1) ) (A.1) 29 Let q$_{2}$ = q − q$_{1}$ , and for each j ∈ { 1 , . . . , m } , let P ( j ) 1 and P ( j ) 2 denote the vectors P$_{1}$ = ( P$_{11}$, . . . , P$_{1}$$_{m}$ ) and P$_{2}$ = ( P$_{21}$, . . . , P$_{2}$$_{m}$ ) with, respectively, P$_{1}$$_{j}$ and P$_{2}$$_{j}$ excluded. For j ∈ { 1 , . . . , m } arbitrary fixed, let R ( j ) 1 ( P ( j ) 1 ) ⊆ { 1 , . . . , j − 1 , j + 1 , . . . , m } be the subset of indices selected along with index j . Note that since the selection rule is valid, this subset is well defined. For any j ∈ { 1 , . . . , m } and given P ( j ) 1 , for i ∈ 1 , . . . , j − 1 , j + 1 , . . . , m we define