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.

翻译：异方差性在同时检验复合零假设的设计有效且有力的程序时，提出了若干方法论挑战。特别是，在此情境下对异方差检验统计量进行标准化或重新缩放的常规做法，可能会严重影响多重检验基础程序的检验功效。此外，当感兴趣推断参数与检验统计量的方差相关时，忽略这种依赖关系的方法可能无法将第一类错误控制在期望水平。我们提出了一种新的异方差调整多重检验（HAMT）程序，该程序避免了通过标准化进行数据缩减，并直接将方差的辅助信息纳入检验过程。我们的方法基于改进的非参数经验贝叶斯反卷积估计量，该估计量为捕捉感兴趣推断参数与检验统计量方差之间的依赖关系提供了实用策略。我们发展了理论，证明HAMT在渐近意义上有效且最优，能够控制错误发现率（FDR）。模拟结果表明，在相同FDR水平下，HAMT在许多设定中显著优于现有程序，并具有实质性的功效提升。该方法通过一个涉及移动游戏应用中活跃用户检测的案例进行了说明。