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- Title
Estimating means of bounded random variables by betting.
- Authors
Waudby-Smith, Ian; Ramdas, Aaditya
- Abstract
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.
- Subjects
MARTINGALES (Mathematics); TEST methods; RANDOM variables
- Publication
Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2024, Vol 86, Issue 1, p1
- ISSN
1369-7412
- Publication type
Article
- DOI
10.1093/jrsssb/qkad009