  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.
