We develop a novel multiple hypothesis testing correction with family-wise error rate (FWER) control that efficiently exploits positive dependencies between potentially correlated statistical hypothesis tests. Our proposed algorithm $\texttt{max-rank}$ is conceptually straight-forward, relying on the use of a $\max$-operator in the rank domain of computed test statistics. We compare our approach to the frequently employed Bonferroni correction, theoretically and empirically demonstrating its superiority over Bonferroni in the case of existing positive dependency, and its equivalence otherwise. Our advantage over Bonferroni increases as the number of tests rises, and we maintain high statistical power whilst ensuring FWER control. We specifically frame our algorithm in the context of parallel permutation testing, a scenario that arises in our primary application of conformal prediction, a recently popularized approach for quantifying uncertainty in complex predictive settings.

翻译：我们开发了一种新型多重假设检验校正方法，可在控制族系错误率（FWER）的同时有效利用潜在相关统计检验之间的正依赖性。提出的$\texttt{max-rank}$算法概念直观，核心在于对计算所得检验统计量的秩域应用$\max$算子。我们将该方法与广泛使用的Bonferroni校正进行理论与实证比较，证明其在存在正依赖性时显著优于Bonferroni，且在无此类依赖性时两者等价。随着检验数量的增加，我们相对于Bonferroni的优势逐步扩大，在确保FWER控制的同时维持高统计功效。我们特别将算法置于并行置换检验的框架中，该场景源于主要应用领域——共形预测，这是近期广受关注的复杂预测场景不确定性量化方法。