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：本文提出一个综合框架，用于调整离散检验统计量以改进其假设检验过程。该调整通过最小化与零假设逼近连续分布之间的Wasserstein距离，解决了离散分布统计显著性组合中的若干基础性难题。相关理论验证了Lancaster中p与均值卡方统计量作为Fisher组合方法的特例。然而，为克服Lancaster检验过程的保守性，我们提出一种更新的零假设逼近分布。该分布通过在适当分布族内进一步最小化与调整后统计量的Wasserstein距离实现。具体到Fisher组合场景，我们提出最优伽马分布替代传统使用的卡方分布。这种新方法能产生渐近一致的检验，显著改善第一类错误控制并增强统计功效。