Asymptotic online FWER control for dependent test statistics

In online multiple testing, an a priori unknown number of hypotheses are tested sequentially, that is, at each time point, a test decision for the current hypothesis has to be made using only the data available so far. Although many powerful test procedures have been developed for online error control in recent years, most of them are designed solely for independent or at most locally dependent test statistics. In this work, we provide a new framework for deriving online multiple test procedures that ensure asymptotical (with respect to the sample size) control of the familywise error rate, regardless of the dependence structure between test statistics. In this context, we give a few concrete examples of such test procedures and discuss their properties. Furthermore, we conduct a simulation study in which the type I error control of these test procedures is also confirmed for a finite sample size, and a gain in power is indicated.