This paper 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. However, in order 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 a proper 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.

翻译：本文提出一个调整离散检验统计量的综合框架，以改进其假设检验流程。该调整通过最小化与零假设近似连续分布之间的Wasserstein距离，解决了离散分布统计显著性组合中固有的若干基本难题。相关理论证明Lancaster中段p值与均值卡方统计量在Fisher组合中可作为特例。然而，为克服Lancaster检验过程的保守性，我们通过进一步在适当分布族内最小化调整后统计量的Wasserstein距离，提出了更新的零假设近似分布。具体在Fisher组合背景下，我们提出用最优伽马分布替代传统卡方分布。这种新方法产生了渐近一致的检验，显著改善了第一类错误控制并提升了统计功效。