Carefree multiple testing with e-processes

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 the FDR, at level $\alpha$ when applied to suprema of $e$-processes. From the same example we see that the FWER is not controlled with averaging, and also closed e-BH does not control the FDR. We show that adjusters can be used to ensure FDR-sup control with e-BH under arbitrary dependence.