  Multiple hypothesis testing is a fundamental problem in high dimensional
inference, with wide applications in many scientific fields. In genome-wide
association studies, tens of thousands of tests are performed simultaneously to
find if any SNPs are associated with some traits and those tests are
correlated. When test statistics are correlated, false discovery control
becomes very challenging under arbitrary dependence. In the current paper, we
propose a novel method based on principal factor approximation, which
successfully subtracts the common dependence and weakens significantly the
correlation structure, to deal with an arbitrary dependence structure. We
derive an approximate expression for false discovery proportion (FDP) in large
scale multiple testing when a common threshold is used and provide a consistent
estimate of realized FDP. This result has important applications in controlling
FDR and FDP. Our estimate of realized FDP compares favorably with Efron
(2007)'s approach, as demonstrated in the simulated examples. Our approach is
further illustrated by some real data applications. We also propose a
dependence-adjusted procedure, which is more powerful than the fixed threshold
procedure.
