  We consider large-scale studies in which it is of interest to test a very
large number of hypotheses, and then to estimate the effect sizes corresponding
to the rejected hypotheses. For instance, this setting arises in the analysis
of gene expression or DNA sequencing data. However, naive estimates of the
effect sizes suffer from selection bias, i.e., some of the largest naive
estimates are large due to chance alone. Many authors have proposed methods to
reduce the effects of selection bias under the assumption that the naive
estimates of the effect sizes are independent. Unfortunately, when the effect
size estimates are dependent, these existing techniques can have very poor
performance, and in practice there will often be dependence. We propose an
estimator that adjusts for selection bias under a recently-proposed frequentist
framework, without the independence assumption. We study some properties of the
proposed estimator, and illustrate that it outperforms past proposals in a
simulation study and on two gene expression data sets.
