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：我们提出了一种新型多重假设检验校正方法，该方法能有效控制族系错误率（FWER），并充分利用潜在相关统计检验之间的正相依性。所提出的算法$\texttt{max-rank}$概念简单直观，其核心在于对计算所得检验统计量的秩域采用$\max$算子。我们将该方法与常用的邦费罗尼校正进行了理论及实证对比，证明在存在正相依性时，该算法明显优于邦费罗尼校正，而在无此条件时则与之等价。随着检验数量的增加，相比邦费罗尼校正的优势持续扩大，同时能在保证FWER控制的前提下维持较高的统计功效。我们特别将算法置于并行置换检验的框架下进行阐述——这一场景源于我们的主要应用领域（共形预测），该方法是近年来在复杂预测场景中量化不确定性的新兴技术。