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

TL;DR

This paper develops an asymptotic theory for optimal sequential multiple testing under generalized error control metrics, proposing procedures that are asymptotically efficient and outperform fixed-sample schemes.

Contribution

It introduces a novel asymptotic optimality framework for sequential multiple testing with generalized error metrics and proposes efficient procedures under independence and LLN conditions.

Findings

Characterized optimal expected sample size asymptotically.

Proposed procedures are asymptotically efficient under various configurations.

Quantified efficiency gains over fixed-sample schemes for i.i.d. data.

Abstract

The sequential multiple testing problem is considered under two generalized error metrics. Under the first one, the probability of at least $k$ mistakes, of any kind, is controlled. Under the second, the probabilities of at least $k_1$ false positives and at least $k_2$ false negatives are simultaneously controlled. For each formulation, the optimal expected sample size is characterized, to a first-order asymptotic approximation as the error probabilities go to 0, and a novel multiple testing procedure is proposed and shown to be asymptotically efficient under every signal configuration. These results are established when the data streams for the various hypotheses are independent and each local log-likelihood ratio statistic satisfies a certain Strong Law of Large Numbers. In the special case of i.i.d. observations in each stream, the gains of the proposed sequential procedures over fixed-sample size schemes are quantified.