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Abstract：
Multiple testing has been a prominent topic in statistical research. Despite extensive work in this area, controlling false discoveries remains a challenging task, especially when the test statistics exhibit dependence. Various methods have been proposed to estimate the false discovery proportion (FDP) under arbitrary dependencies among the test statistics. One key approach is to transform arbitrary dependence into weak dependence and subsequently establish the strong consistency of FDP and false discovery rate under weak dependence. As a result, FDPs converge to the same asymptotic limit within the framework of weak dependence. However, we have observed that the asymptotic variance of FDP can be significantly influenced by the dependence structure of the test statistics, even when they exhibit only weak dependence. Quantifying this variability is of great practical importance, as it serves as an indicator of the quality of FDP estimation from the data. To the best of our knowledge, there is limited research on this aspect in the literature. In this paper, we aim to fill in this gap by quantifying the variation of FDP, assuming that the test statistics exhibit weak dependence and follow normal distributions. We begin by deriving the asymptotic expansion of the FDP and subsequently investigate how the asymptotic variance of the FDP is influenced by different dependence structures. Based on the insights gained from this study, we recommend that in multiple testing procedures utilizing FDP, reporting both the mean and variance estimates of FDP can provide a more comprehensive assessment of the study\'s outcomes.
摘要：
多重检验一直是统计研究中的一个突出课题。尽管在这方面做了大量的工作，控制错误发现仍然是一项具有挑战性的任务，特别是当检验统计量表现出依赖性时。已经提出了各种方法来估计在测试统计量之间的任意依赖性下的错误发现比例（FDP）。一种关键方法是将任意依赖转化为弱依赖，并随后建立FDP的强一致性和弱依赖下的错误发现率。因此,FDP在弱依赖框架内收敛到相同的渐近极限。然而,我们已经观察到，FDP的渐近方差可以显著影响的依赖结构的检验统计，即使它们只表现出微弱的依赖性。量化这种可变性具有非常重要的实际意义，因为它可以作为从数据中评估FDP质量的指标。据我们所知,文献中对这方面的研究有限。在本文中,我们的目标是通过量化FDP的变化来填补这一空白，假设检验统计量表现出弱依赖性，服从正态分布。我们首先推导FDP的渐近展开，然后研究FDP的渐近方差如何受到不同依赖结构的影响。基于从这项研究中获得的见解，我们建议在使用FDP的多个测试程序中，报告FDP的均值和方差估计值可以为研究结果提供更全面的评估.
