Estimating means of bounded random variables by betting
This provides improved statistical bounds for fundamental problems in data analysis, benefiting researchers and practitioners in fields requiring precise mean estimation, though it is an incremental advancement building on prior methods like Chernoff.
The paper tackles the problem of estimating unknown means from bounded observations by deriving confidence intervals and sequences that adapt to unknown variance, achieving a new state-of-the-art with empirical performance vastly outperforming existing methods like Hoeffding and empirical Bernstein inequalities.
This paper derives confidence intervals (CI) and time-uniform confidence sequences (CS) for the classical problem of estimating an unknown mean from bounded observations. We present a general approach for deriving concentration bounds, that can be seen as a generalization and improvement of the celebrated Chernoff method. At its heart, it is based on a class of composite nonnegative martingales, with strong connections to testing by betting and the method of mixtures. We show how to extend these ideas to sampling without replacement, another heavily studied problem. In all cases, our bounds are adaptive to the unknown variance, and empirically vastly outperform existing approaches based on Hoeffding or empirical Bernstein inequalities and their recent supermartingale generalizations. In short, we establish a new state-of-the-art for four fundamental problems: CSs and CIs for bounded means, when sampling with and without replacement.