Combining p-values via averaging

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. An old result by R\"uschendorf and, independently, Meng 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). Based on more recent developments in mathematical finance, specifically, robust risk aggregation techniques, we show that $K$ p-values can be combined by scaling up their geometric mean by a factor of $e$ (for all $K$) and by scaling up their harmonic mean by a factor of $\ln K$ (asymptotically as $K\to\infty$). These and other results lead to a generalized version of the Bonferroni-Holm method. A simulation study compares the performance of various averaging methods.