We discover a connection between the Benjamini-Hochberg (BH) procedure and the recently proposed e-BH procedure [Wang and Ramdas, 2022] with a suitably defined set of e-values. This insight extends to a generalized version of the BH procedure and the model-free multiple testing procedure in Barber and Cand\`es [2015] (BC) with a general form of rejection rules. The connection provides an effective way of developing new multiple testing procedures by aggregating or assembling e-values resulting from the BH and BC procedures and their use in different subsets of the data. In particular, we propose new multiple testing methodologies in three applications, including a hybrid approach that integrates the BH and BC procedures, a multiple testing procedure aimed at ensuring a new notion of fairness by controlling both the group-wise and overall false discovery rates (FDR), and a structure adaptive multiple testing procedure that can incorporate external covariate information to boost detection power. One notable feature of the proposed methods is that we use a data-dependent approach for assigning weights to e-values, significantly enhancing the efficiency of the resulting e-BH procedure. The construction of the weights is non-trivial and is motivated by the leave-one-out analysis for the BH and BC procedures. In theory, we prove that the proposed e-BH procedures with data-dependent weights in the three applications ensure finite sample FDR control. Furthermore, we demonstrate the efficiency of the proposed methods through numerical studies in the three applications.

翻译：我们发现Benjamini-Hochberg (BH)程序与新近提出的e-BH程序[Wang and Ramdas, 2022]之间，在适当定义一组e-值的情况下存在联系。这一洞见可推广至BH程序的广义版本，以及Barber和Candès[2015] (BC)中采用一般形式拒绝规则的无模型多重检验程序。该联系通过聚合或组合源自BH和BC程序及其在不同数据子集中应用所产生的e-值，为开发新的多重检验程序提供了有效途径。具体而言，我们在三个应用中提出了新的多重检验方法：一种整合BH和BC程序的混合方法；一种旨在通过同时控制组间和整体错误发现率（FDR）来确保新型公平性概念的多重检验程序；以及一种可整合外部协变量信息以提升检测能力的结构自适应多重检验程序。所提方法的显著特征在于采用数据依赖方式为e-值分配权重，显著提升了所得e-BH程序的效率。该权重的构造具有非平凡性，其动机源于BH和BC程序的留一分析。理论上，我们证明了在三个应用中提出的带数据依赖权重的e-BH程序能确保有限样本FDR控制。此外，我们通过三个应用中的数值研究验证了所提方法的有效性。