  The highly influential two-group model in testing a large number of
statistical hypotheses assumes that the test statistics are drawn independently
from a mixture of a high probability null distribution and a low probability
alternative. Optimal control of the marginal false discovery rate (mFDR), in
the sense that it provides maximal power (expected true discoveries) subject to
mFDR control, is known to be achieved by thresholding the local false discovery
rate (locFDR), i.e., the probability of the hypothesis being null given the set
of test statistics, with a fixed threshold. We address the challenge of
controlling optimally the popular false discovery rate (FDR) or positive FDR
(pFDR) rather than mFDR in the general two-group model, which also allows for
dependence between the test statistics. These criteria are less conservative
than the mFDR criterion, so they make more rejections in expectation. We derive
their optimal multiple testing (OMT) policies, which turn out to be
thresholding the locFDR with a threshold that is a function of the entire set
of statistics. We develop an efficient algorithm for finding these policies,
and use it for problems with thousands of hypotheses. We illustrate these
procedures on gene expression studies.
