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 online procedures have been proposed, but none of them are generalizations of the Benjamini-Hochberg (BH) procedure based on p-values, or of the e-BH procedure 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. We show that the FDR guarantees for BH (resp. e-BH) and online BH (resp. online e-BH) are identical under positive, negative or arbitrary dependence, at fixed and stopping times. Further, the online BH (resp. online e-BH) rule recovers the BH (resp. e-BH) rule as a special case when the number of hypotheses is known to be fixed. Of independent interest, our proof techniques also allow us to prove that numerous existing online procedures, which were 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，假设逐一到达，每个时刻我们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。已有多种在线检验方法被提出，但尚无一种能基于p值推广经典的Benjamini-Hochberg（BH）过程，或基于e值推广e-BH过程。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。我们证明这种放宽使得我们可以定义——并通过充分论证说明其合理性——BH过程与e-BH过程在在线场景下自然且恰当的扩展形式。我们证明，无论p值（或e值）之间存在正相关、负相关或任意依赖关系，在固定时刻或停止时刻下，BH（对应e-BH）过程与其在线版本具有完全相同的错误发现率（FDR）控制保证。进一步地，当假设总数已知且固定时，在线BH（对应在线e-BH）规则可退化为标准的BH（对应e-BH）规则。作为独立贡献，我们的证明技术还允许我们证实：众多已知能在固定时刻控制FDR的现有在线方法，同样能在停止时刻控制FDR。