This paper proposes general methods for the problem of multiple testing of a single hypothesis, with a standard goal of combining a number of |$p$|-values without making any assumptions about their dependence structure. A result by Rüschendorf (1982) and, independently, Meng (1993) implies that the |$p$|-values can be combined by scaling up their arithmetic mean by a factor of 2, and no smaller factor is sufficient in general. A similar result by Mattner about the geometric mean replaces 2 by e. Based on more recent developments in mathematical finance, specifically, robust risk aggregation techniques, we extend these results to generalized means; in particular, we show that |$K$||$p$|-values can be combined by scaling up their harmonic mean by a factor of |$\log K$| asymptotically as |$K$| tends to infinity. This leads to a generalized version of the Bonferroni–Holm procedure. We also explore methods using weighted averages of |$p$|-values. Finally, we discuss the efficiency of various methods of combining |$p$|-values and how to choose a suitable method in light of data and prior information.

中文翻译：

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通过求平均值合并p值

本文提出了针对单个假设的多重检验问题的一般方法，其标准目标是合并多个| $ p $ |。-值，而无需对其依赖结构进行任何假设。结果由Rüschendorf（1982），并独立地 Meng（1993）暗示| $ p $ | 值可以通过将其算术平均值按比例放大2来组合，通常没有较小的比例足够。Mattner关于几何平均值的类似结果将2替换为e。基于数学金融的最新发展，特别是稳健的风险汇总技术，我们将这些结果扩展到广义均值。特别是，我们表明| $ K $ | | $ p $ | 值可以通过将其谐波均值按| $ \ log K $ |的比例进行组合 渐近地为| $ K $ | 趋于无穷大。这导致了Bonferroni–Holm程序的广义版本。我们还将探索使用| $ p $ |的加权平均值的方法。值。最后，我们讨论了| $ p $ |合并的各种方法的效率。值以及如何根据数据和先验信息选择合适的方法。