Considering the knockoff-based multiple testing framework of Barber and Candès [2015], we revisit the method of Sarkar and Tang [2022] and identify it as a specific case of an un-normalized e-value weighted Benjamini-Hochberg procedure. Building on this insight, we extend the method to use bounded p-to-e calibrators that enable more refined and flexible weight assignments. Our approach generalizes the method of Sarkar and Tang [2022], which emerges as a special case corresponding to an extreme calibrator. Within this framework, we propose three procedures: an e-value weighted Benjamini-Hochberg method, its adaptive extension using an estimate of the proportion of true null hypotheses, and an adaptive weighted Benjamini-Hochberg method. We establish control of the false discovery rate (FDR) for the proposed methods. While we do not formally prove that the proposed methods outperform those of Barber and Candès [2015] and Sarkar and Tang [2022], simulation studies and real-data analysis demonstrate large and consistent improvement over the latter in all cases, and better performance than the knockoff method in scenarios with low target FDR, a small number of signals, and weak signal strength. Simulation studies and a real-data application in HIV-1 drug resistance analysis demonstrate strong finite sample FDR control and exhibit improved, or at least competitive, power relative to the aforementioned methods.

翻译：针对Barber与Candès [2015]提出的基于knockoff的多重检验框架，我们重新审视了Sarkar与Tang [2022]的方法，并将其识别为未归一化e值加权Benjamini-Hochberg过程的一个特例。基于这一认识，我们将该方法扩展至使用有界p值到e值的校准函数，从而实现更精细和灵活的权重分配。我们的方法推广了Sarkar与Tang [2022]的方法——后者对应于极端校准函数的特殊情况。在此框架下，我们提出了三种方法：e值加权Benjamini-Hochberg方法、使用真实零假设比例估计的自适应扩展方法，以及自适应加权Benjamini-Hochberg方法。我们证明了所提方法对错误发现率（FDR）的控制能力。虽然未从形式上证明所提方法优于Barber与Candès [2015]和Sarkar与Tang [2022]的方法，但模拟研究和实际数据分析表明：在所有情况下，新方法相对后者均表现出显著且一致的改进；在目标FDR较低、信号数量较少且信号强度较弱的情境下，其性能也优于knockoff方法。HIV-1耐药性分析中的模拟研究和实际数据应用表明，所提方法具有强大的有限样本FDR控制能力，并且相较于前述方法展现出提升的——或至少具有竞争力的——检验功效。