Optimality

0.025
0.02
0.015
0.01
0.005
On the false discovery proportion 45
α i ’’
0
0 10 20 30 40 50
i
60 70 80 90 100
η
x 10−3
4
3
2
1
i ’
0
0 10 20 30 40 50
i
60 70 80 90 100
8
6
4
2
α i ’’/η i ’
0
0 10 20 30 40 50
i
60 70 80 90 100
Fig 1. Stepdown Constants for s = 100, γ = .1, and α = .05.
that of independence of p-values used by Benjamini and Liu [2].
Theorem 4.1. For testing Hi : P ∈ ωi, i = 1, . . . , s, suppose ˆpi satisfies (2.1).
Consider the stepdown procedure with constants
(4.1) α ∗ i = min{
sα
,1}
(s−i+1) 2
and assume the condition (3.5). Then, FDR≤α.
Proof. First note that if|I| = 0, then FDR = 0. Second, if|I| = s, then FDR =
P{ˆp (1)≤ α ∗ 1}≤ � s
i=1 P{ˆpi≤ α ∗ 1}≤sα ∗ 1 = α.
Now suppose that 0 α ∗ 1). Define t to
be the total number of true hypotheses rejected by the stepdown procedure and
f to be the total number of false hypotheses rejected by the stepdown procedure.