A minimum Wasserstein distance approach to Fisher's combination of independent, discrete p‐values

This article introduces a comprehensive framework to adjust a discrete test statistic for improving its hypothesis testing procedure. The adjustment minimizes the Wasserstein distance to a null‐approximating continuous distribution, tackling some fundamental challenges inherent in combining statistical significances derived from discrete distributions. The related theory justifies Lancaster's mid‐p and mean‐value chi‐squared statistics for Fisher's combination as special cases. To counter the conservative nature of Lancaster's testing procedures, we propose an updated null‐approximating distribution. It is achieved by further minimizing the Wasserstein distance to the adjusted statistics within an appropriate distribution family. Specifically, in the context of Fisher's combination, we propose an optimal gamma distribution as a substitute for the traditionally used chi‐squared distribution. This new approach yields an asymptotically consistent test that significantly improves Type I error control and enhances statistical power.