An Upper Confidence Bound Approach to Estimating the Maximum Mean
This work addresses a fundamental statistical estimation problem with applications in areas like clinical trials, though it is incremental as it builds on existing methods.
The paper tackles the problem of estimating the maximum mean using an upper confidence bound approach with adaptive sampling, proposing a new largest-size average estimator that shows faster bias decay than the existing grand average estimator, as validated by statistical guarantees and numerical examples.
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.