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.

翻译：最大均值的估计在实践中有多种应用。本文研究采用上置信界方法估计最大均值，其中采样预算被自适应地分配给系统之一。我们深入研究了现有的总平均估计量，并提出了一种新的最大样本量平均估计量。具体而言，我们为两种估计量建立了统计保证，包括强相合性、渐近均方误差和中心极限定理，这些结果在现有文献中均属首次提出。我们证明最大样本量平均估计量优于总平均估计量，因为当样本量增加时，前者的偏差衰减速率远快于后者。利用中心极限定理，我们进一步构建了最大均值的渐近有效置信区间，并提出了一种适用于临床试验中多重比较问题的单一假设检验方法。通过数值算例，验证了所得点估计与区间估计以及所提单一假设检验方法的统计效率。