Often in multiple testing, the hypotheses appear in non-overlapping blockswith the associated $p$-values exhibiting dependence within but not betweenblocks. We consider adapting the Benjamini-Hochberg method for controlling thefalse discovery rate (FDR) and the Bonferroni method for controlling thefamilywise error rate (FWER) to such dependence structure without losing theirultimate controls over the FDR and FWER, respectively, in a non-asymptoticsetting. We present variants of conventional adaptive Benjamini-Hochberg andBonferroni methods with proofs of their respective controls over the FDR andFWER. Numerical evidence is presented to show that these new adaptive methodscan capture the present dependence structure more effectively than thecorresponding conventional adaptive methods. This paper offers a solution tothe open problem of constructing adaptive FDR and FWER controlling methodsunder dependence in a non-asymptotic setting and providing real improvementsover the corresponding non-adaptive ones.
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机译：通常在多个测试中，假设出现在非重叠块中，关联的$ p $ -values展示内部依赖性但未介于介绍之间。我们考虑调整Benjamini-Hochberg方法，用于控制FALSE发现率（FDR）和Bonferroni方法，用于控制这种依赖结构的自由误差率（FWER），而不会分别在非渐关节功能中通过FDR和FWER对FDR和FWER进行分析。我们呈现常规自适应Benjamini-Hochberg Andbonferroni方法的变体，并通过其各自的FDR和FFWer的证据证明。提出了数值证据表明，这些新的自适应方法扫描比对应的传统自适应方法更有效地捕获本依赖性结构。本文提供了一个解决方案，使得构建自适应FDR和FWER控制方法在非渐近设置中的依赖性和提供实际改进的不适应非自适应的问题。
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