  Heteroskedasticity poses several methodological challenges in designing valid
and powerful procedures for simultaneous testing of composite null hypotheses.
In particular, the conventional practice of standardizing or re-scaling
heteroskedastic test statistics in this setting may severely affect the power
of the underlying multiple testing procedure. Additionally, when the
inferential parameter of interest is correlated with the variance of the test
statistic, methods that ignore this dependence may fail to control the type I
error at the desired level. We propose a new Heteroskedasticity Adjusted
Multiple Testing (HAMT) procedure that avoids data reduction by
standardization, and directly incorporates the side information from the
variances into the testing procedure. Our approach relies on an improved
nonparametric empirical Bayes deconvolution estimator that offers a practical
strategy for capturing the dependence between the inferential parameter of
interest and the variance of the test statistic. We develop theory to show that
HAMT is asymptotically valid and optimal for FDR control. Simulation results
demonstrate that HAMT outperforms existing procedures with substantial power
gain across many settings at the same FDR level. The method is illustrated on
an application involving the detection of engaged users on a mobile game app.
