An Upper Confidence Bound Approach to Estimating the Maximum Mean

TL;DR

This paper introduces a new estimator for the maximum mean using an upper confidence bound approach, providing statistical guarantees and demonstrating improved bias decay over existing methods, with applications in clinical trials.

Contribution

The paper proposes a novel largest-size average (LSA) estimator for maximum mean estimation, with proven statistical properties and advantages over the grand average (GA) estimator.

Findings

LSA estimator has faster bias decay than GA.

Constructs asymptotically valid confidence intervals for maximum mean.

Demonstrates statistical efficiency through numerical examples.

Abstract

Estimating the maximum mean finds a variety of applications in practice. In this paper, we study estimation of the maximum mean using an upper confidence bound (UCB) approach where the sampling budget is adaptively allocated to one of the systems. We study in depth the existing grand average (GA) estimator, and propose a new largest-size average (LSA) estimator. Specifically, we establish statistical guarantees, including strong consistency, asymptotic mean squared errors, and central limit theorems (CLTs) for both estimators, which are new to the literature. We show that LSA is preferable over GA, as the bias of the former decays at a rate much faster than that of the latter when sample size increases. By using the CLTs, we further construct asymptotically valid confidence intervals for the maximum mean, and propose a single hypothesis test for a multiple comparison problem with application to clinical trials. Statistical efficiency of the resulting point and interval estimates and the proposed single hypothesis test is demonstrated via numerical examples.