Optimality

68 C. Cheng
root MSE
bias
0.0 0.04 0.08 0.12
-0.15 -0.05 0.05 0.15
Models 1,3,5,9; sigma=3
0.82 0.86 0.90 0.94 0.98
true pi0
Models 1,3,5,9; sigma=3
0.82 0.86 0.90 0.94 0.98
true pi0
root MSE
bias
0.0 0.015 0.030 0.045 0.060
-0.15 -0.05 0.05 0.15
Models 2,4,6,10; sigma=1
0.82 0.86 0.90 0.94 0.98
true pi0
Models 2,4,6,10; sigma=1
0.82 0.86 0.90 0.94 0.98
true pi0
Fig 3. Root MSE and bias of the π0 estimators by Benjamini and Hochberg [3] (circle), Storey
et al. [31] (triangle), and (3.1) (diamond)
and the bias of the proposed and the Benjamini and Hochberg’s estimators are
reduced substantially while the small downward bias of Storey’s bootstrap estimator
remains. Overall the proposed estimator (3.1) outperforms the other two estimators
in terms of MSE and bias.
Next, operating characteristics of the adaptive FDR control ([3]) and q-value
FDR control ([31]) at the 1%, 5%, 10%, 15%, 20%, 30%, 40%, 60%, and 70%
levels, the criteria API (i.e., the HT(�α ∗ cal ) procedure) and Ip ([7]), are simulated
and compared. The performance measures are the estimated FDR (�FDR) and the
estimated false nondiscovery proportion ( FNDP) � defined as follows. Let m1 be the
number of true alternative hypotheses according to the simulation model, let Rl be
the total number of rejections in simulation trial l, and let Sl be the number of
correct rejections. Define
�FDR = 1
1000
FNDP � = 1
1000
� 1000
l=1 I(Rl > 0)(Rl− Sl) � Rl
�1000 l=1 (m1− Sl) � m1,
where I(·) is the indicator function. These are the Monte Carlo estimators of the