Sequential multiple testing with generalized error control: an asymptotic optimality theory

The multiple testing problem is considered under two different error metrics, when the data for the various hypotheses are collected sequentially in independent streams. In the first one, the probability of making at least $k$ mistakes, of any kind, is controlled. In the second, the probabilities of at least $k_1$ false positives and at least $k_2$ false negatives are simultaneously controlled below two arbitrary levels. For each formulation, we characterize the optimal expected sample size to a first-order asymptotic approximation as the error probabilities vanish (at arbitrary rates). More importantly, for each formulation we propose a novel, feasible sequential multiple testing procedure that achieves the optimal asymptotic performance under every possible signal configuration. These asymptotic optimality results are established under weak distributional assumptions which hold beyond the case of i.i.d. observations in the streams.