Stepwise multiple testing procedures have attracted several statisticians for decades and are also quite popular with statistics users because of their technical simplicity. The Bonferroni procedure has been one of the earliest and most prominent testing rules for controlling the familywise error rate (FWER). A recent article established that the FWER for the Bonferroni method asymptotically (i.e., when the number of hypotheses becomes arbitrarily large) approaches zero under any positively equicorrelated multivariate normal framework. However, similar results for the limiting behaviors of FWER of general stepwise procedures are nonexistent. The present work addresses this gap in a unified manner by studying the limiting behaviors of the FWER of several stepwise testing rules for correlated normal setups. Specifically, we show that the limiting FWER approaches zero for any step-down rule (e.g., Holm's method) provided the infimum of the correlations is strictly positive. We also establish similar limiting zero results on FWER of other popular multiple testing rules, e.g., Hochberg's and Hommel's procedures. We then extend these results to any configuration of true and false null hypotheses. It turns out that, within our chosen asymptotic framework, the Benjamini-Hochberg method can hold the FWER at a strictly positive level asymptotically under the equicorrelated normality. We finally discuss the limiting powers of various procedures.

翻译：逐步多重检验程序因其技术简便性，数十年来吸引了众多统计学研究者，并在统计用户中广受欢迎。Bonferroni 程序是最早且最突出的用于控制族系误差率的检验规则之一。近期一篇文章证明，在任意正等相关的多元正态框架下，Bonferroni 方法的 FWER 渐近地（即当假设数量趋于无穷大时）趋近于零。然而，关于一般逐步检验程序 FWER 极限行为的类似结果尚不存在。本研究通过统一研究相关正态设定下多种逐步检验规则的 FWER 极限行为，填补了此空白。具体而言，我们证明只要相关系数的下确界严格为正，任何逐步向下规则（如 Holm 方法）的极限 FWER 均趋近于零。我们同样建立了其他流行多重检验规则（如 Hochberg 和 Hommel 程序）FWER 的类似极限零结果。随后，我们将这些结果推广到真和伪零假设的任意配置。结果表明，在我们选择的渐近框架内，Benjamini-Hochberg 方法在等相关正态性下可渐近地将 FWER 维持在严格正值水平。最后，我们讨论了各程序的极限势。