Handling multiplicity without losing much power has been a persistent challenge in various fields that often face the necessity of managing numerous statistical tests simultaneously. Recently, $p$-value combination methods based on heavy-tailed distributions, such as a Cauchy distribution, have received much attention for their ability to handle multiplicity without the prescribed knowledge of the dependence structure. This paper delves into these types of $p$-value combinations through the lens of extreme value theory. Distributions with regularly varying tails, a subclass of heavy tail distributions, are found to be useful in constructing such $p$-value combinations. Three $p$-value combination statistics (sum, max cumulative sum, and max) are introduced, of which left tail probabilities are shown to be approximately uniform when the global null is true. The primary objective of this paper is to bridge the gap between current developments in $p$-value combination methods and the literature on extreme value theory, while also offering guidance on selecting the calibrator and its associated parameters.

翻译：在多领域研究中，同时管理大量统计检验时，如何在控制多重性的同时保持统计功效始终是持续挑战。近年来，基于重尾分布（如柯西分布）的$p$值组合方法因无需预先了解依赖结构即可处理多重性而备受关注。本文从极值理论视角深入探讨此类$p$值组合方法。研究发现，具有正则变化尾部的分布（重尾分布的子类）在构建此类$p$值组合中具有实用价值。本文引入三种$p$值组合统计量（和、累积最大值和、最大值），在全局零假设成立条件下，其左尾概率近似服从均匀分布。本文的核心目标在于弥合当前$p$值组合方法的发展与极值理论文献之间的鸿沟，同时为校准器及其参数的选择提供指导性建议。