In online multiple testing, the hypotheses arrive one by one, and at each time we must immediately reject or accept the current hypothesis solely based on the data and hypotheses observed so far. Many procedures have been proposed, but none of them are online generalizations of the Benjamini-Hochberg (BH) procedure based on p-values, or of the e-BH procedures that uses e-values. In this paper, we consider a relaxed problem setup that allows the current hypothesis to be rejected at any later step. We show that this relaxation allows us to define -- what we justify extensively to be -- the natural and appropriate online extension of the BH and e-BH procedures. Analogous to e-BH, online e-BH controls the FDR under arbitrary dependence (even at stopping times). Like for e-BH, we show how to boost the power of online e-BH under other dependence assumptions like positive or local dependence. BH and online BH have identical FDR guarantees at fixed times under positive, negative or arbitrary dependence. Further, we prove that online BH has a slightly inflated FDR control at data-adaptive stopping times under weak positive and negative dependence. Based on the same proof techniques, we prove that numerous existing online procedures, which were previously only known to control the FDR at fixed times, also control the FDR at stopping times.

翻译：在线多重假设检验中，假设按序到达，每个时刻必须仅基于当前已观测到的数据和假设立即对当前假设做出拒绝或接受的判定。已有多种方法被提出，但尚未出现基于p值的Benjamini-Hochberg（BH）方法或使用e值的e-BH方法的在线泛化版本。本文考虑一种放宽的问题设定，允许当前假设在后续任意时刻被拒绝。我们证明这种放宽条件使得能够定义——经充分论证为——BH与e-BH方法自然且恰当的在线扩展。类比于e-BH，在线e-BH能在任意依赖性（甚至在停止时间）下控制FDR。与e-BH类似，我们展示了如何在正依赖或局部依赖等其他依赖假设下提升在线e-BH的检验功效。在正依赖、负依赖或任意依赖条件下，BH与在线BH在固定时间点具有相同的FDR保证。进一步，我们证明在线BH在弱正依赖与弱负依赖条件下，在数据自适应的停止时间具有轻微膨胀的FDR控制。基于相同的证明技术，我们证实众多现有在线方法——这些方法先前仅被证明能在固定时间控制FDR——同样能在停止时间控制FDR。