  Efforts to develop more efficient multiple hypothesis testing procedures for
false discovery rate (FDR) control have focused on incorporating an estimate of
the proportion of true null hypotheses (such procedures are called adaptive) or
exploiting heterogeneity across tests via some optimal weighting scheme. This
paper combines these approaches using a weighted adaptive multiple decision
function (WAMDF) framework. Optimal weights for a flexible random effects model
are derived and a WAMDF that controls the FDR for arbitrary weighting schemes
when test statistics are independent under the null hypotheses is given.
Asymptotic and numerical assessment reveals that, under weak dependence, the
proposed WAMDFs provide more efficient FDR control even if optimal weights are
misspecified. The robustness and flexibility of the proposed methodology
facilitates the development of more efficient, yet practical, FDR procedures
for heterogeneous data. To illustrate, two different weighted adaptive FDR
methods for heterogeneous sample sizes are developed and applied to data.
