Multiple testing adjustments, such as the Benjamini & Hochberg (1995) step-up procedure for controlling the false discovery rate (FDR), are typically applied to families of tests that control significance level in the classical sense: for each individual test, the probability of false rejection is no greater than the nominal level. In this paper, we consider tests that satisfy only a weaker notion of significance level control, in which the probability of false rejection need only be controlled on average over the hypotheses. We find that the Benjamini & Hochberg (1995) step-up procedure still controls FDR in the asymptotic regime with many weakly dependent p-values and an increasing number of rejections, and that certain adjustments for dependent p-values such as the Benjamini & Yekutieli (2001) procedure continue to yield FDR control in finite samples. Our results open the door to FDR controlling procedures in nonparametric and high dimensional settings where weakening the notion of inference may allow for power improvements.

翻译：多重检验调整方法（例如 Benjamini & Hochberg (1995) 用于控制虚假发现率（FDR）的逐步递增程序）通常应用于在经典意义上控制显著性水平的检验族：对于每个单独检验，错误拒绝的概率不超过名义水平。本文考虑仅满足较弱显著性水平控制概念的检验，其中错误拒绝的概率只需在假设上取平均控制。我们发现在渐近框架下，对于大量弱相依的p值及递增的拒绝次数，Benjamini & Hochberg (1995) 逐步递增程序仍能控制FDR；而针对相依p值的某些调整方法（如 Benjamini & Yekutieli (2001) 程序）在有限样本中仍能产生FDR控制。我们的结果为在非参数和高维设定中应用FDR控制程序开辟了道路，在此类设定中放宽推断概念可能允许提升检验效能。