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An Upper Confidence Bound Approach to Estimating the Maximum Mean

Zhang Kun, Liu Guangwu, Shi Wen

TL;DR

This work tackles estimating the maximum mean $\mu^* = \max_k \mu_k$ across $K$ stochastic systems under adaptive sampling via a generalized UCB policy. It introduces two estimators, the Grand Average $\widetilde{M}_n$ and the Largest-Size Average $M_{I^*_n}$ (LSA), and establishes strong consistency, $\text{MSE}=O(1/n)$ or $O(\nu_n/n^{3/2})$ biases, and central limit theorems for both, enabling asymptotically valid CIs. The LSA estimator is shown to have faster bias decay $O(\nu_n/n^{3/2})$ than GA's $O(\nu_n/n)$, and both support a single, powerful hypothesis test for maximal-mean comparisons in clinical trials and related settings. Numerical experiments on coherent risk measures, clinical trials, and call-center robustness confirm that LSA offers superior finite-sample performance and tighter inference while maintaining asymptotic guarantees.

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.

An Upper Confidence Bound Approach to Estimating the Maximum Mean

TL;DR

This work tackles estimating the maximum mean across stochastic systems under adaptive sampling via a generalized UCB policy. It introduces two estimators, the Grand Average and the Largest-Size Average (LSA), and establishes strong consistency, or biases, and central limit theorems for both, enabling asymptotically valid CIs. The LSA estimator is shown to have faster bias decay than GA's , and both support a single, powerful hypothesis test for maximal-mean comparisons in clinical trials and related settings. Numerical experiments on coherent risk measures, clinical trials, and call-center robustness confirm that LSA offers superior finite-sample performance and tighter inference while maintaining asymptotic guarantees.

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.
Paper Structure (28 sections, 15 theorems, 162 equations, 5 figures, 5 tables, 2 algorithms)

This paper contains 28 sections, 15 theorems, 162 equations, 5 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem and Backgrounds
  3. An Upper Confidence Bound Approach
  4. Grand Average (GA) Estimator
  5. Largest-Size Average (LSA) Estimator
  6. Hypothesis Testing using Maximum Mean Estimators
  7. Theoretical Analysis
  8. GA Estimator
  9. LSA Estimator
  10. Numerical Study
  11. Implementation Issues
  12. Maximum of $K$ Normal Means
  13. Simulated CIs for Coherent Risk Measures
  14. Hypothesis Testing in Clinical Trials
  15. Robust Analysis of Stochastic Simulation: A Call Center Case
  16. ...and 13 more sections

Key Result

Proposition 1

Suppose that Assumptions ass:subGaussian and ass:unique hold, and $\nu_n\geq4\bar{\gamma}^2\log n$. Then, for $k\neq k^*$ and any positive integer $p$, where $\Delta_k>0$ for $k\neq k^*$ due to Assumption ass:unique, and the notation $O(\cdot)$ means that $\limsup_{n\rightarrow\infty} a_n/b_n\le C$ for some constant $C$ if $a_n=O(b_n)$.

Figures (5)

  • Figure 1: (Color online) Convergence of estimated absolute biases.
  • Figure 2: (Color online) Convergence rate of estimated MSEs.
  • Figure 3: (Color online) Observed coverage probabilities of the 90% confidence intervals with respect to different sample sizes.
  • Figure 4: (Color online) Comparison of CIs
  • Figure 5: Logic flow of the call center of an anonymous bank

Theorems & Definitions (18)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 2
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • ...and 8 more