Estimating means of bounded random variables by betting

• 成果类型: Article
• 署名作者: Waudby-Smith, Ian; Ramdas, Aaditya
• 署名单位: Carnegie Mellon University; Carnegie Mellon University; Carnegie Mellon University
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1093/jrsssb/qkad009
• 发表日期: 2024
• 关键词: probability-inequalities; time-uniform; INFORMATION; strategy; SUM; LAW

摘要

We derive confidence intervals (CIs) and confidence sequences (CSs) for the classical problem of estimating a bounded mean. Our approach generalizes and improves on the celebrated Chernoff method, yielding the best closed-form empirical-Bernstein CSs and CIs (converging exactly to the oracle Bernstein width) as well as non-closed-form betting CSs and CIs. Our method combines new composite nonnegative (super)martingales with Ville's maximal inequality, with strong connections to testing by betting and the method of mixtures. We also show how these ideas can be extended to sampling without replacement. In all cases, our bounds are adaptive to the unknown variance, and empirically vastly outperform prior approaches, establishing a new state-of-the-art for four fundamental problems: CSs and CIs for bounded means, when sampling with and without replacement.