E-processes enable hypothesis testing with ongoing data collection while maintaining Type I error control. However, when testing multiple hypotheses simultaneously, current $e$-value based multiple testing methods such as e-BH are not invariant to the order in which data are gathered for the different $e$-processes. This can lead to undesirable situations, e.g., where a hypothesis rejected at time $t$ is no longer rejected at time $t+1$ after choosing to gather more data for one or more $e$-processes unrelated to that hypothesis. We argue that multiple testing methods should always work with suprema of $e$-processes. We provide an example to illustrate that e-BH does not control this FDR at level $\alpha$ when applied to suprema of $e$-processes. We show that adjusters can be used to ensure FDR-sup control with e-BH under arbitrary dependence.

翻译：e-过程支持在持续收集数据的同时进行假设检验，同时保持第一类错误控制。然而，当同时检验多个假设时，当前基于e值的多重检验方法（如e-BH）对不同e过程的数据收集顺序不具有不变性。这可能导致不良情况，例如在时间t被拒绝的假设，在决定为与该假设无关的一个或多个e过程收集更多数据后，于时间t+1不再被拒绝。我们认为多重检验方法应始终基于e过程的上确界进行。我们提供一个示例说明当e-BH应用于e过程的上确界时，无法在水平α上控制FDR。我们证明在任意依赖关系下，可通过调整器确保e-BH实现FDR-上确界控制。