Weighted Asymptotically Optimal Sequential Testing

TL;DR

This paper introduces weighted sequential testing procedures that incorporate prior information, achieving asymptotic optimality and strong error control even in high-dimensional and random-weight settings.

Contribution

It develops weighted log-likelihood ratio methods and proves their asymptotic optimality and robustness in complex, high-dimensional scenarios.

Findings

Weighted procedures control family-wise error rate effectively.

Expected stopping times reach the theoretical lower bound.

Methods are robust with respect to high-dimensional and random weights.

Abstract

This paper develops a framework for incorporating prior information into sequential multiple testing procedures while maintaining asymptotic optimality. We define a weighted log-likelihood ratio (WLLR) as an additive modification of the standard LLR and use it to construct two new sequential tests: the Weighted Gap and Weighted Gap-Intersection procedures. We prove that both procedures provide strong control of the family-wise error rate. Our main theoretical contribution is to show that these weighted procedures are asymptotically optimal; their expected stopping times achieve the theoretical lower bound as the error probabilities vanish. This first-order optimality is shown to be robust, holding in high-dimensional regimes where the number of null hypotheses grows and in settings with random weights, provided that mild, interpretable conditions on the weight distribution are met.