Constructing nonasymptotic confidence intervals (CIs) for the mean of a univariate distribution from independent and identically distributed (i.i.d.) observations is a fundamental task in statistics. For bounded observations, a classical nonparametric approach proceeds by inverting standard concentration bounds, such as Hoeffding's or Bernstein's inequalities. Recently, an alternative betting-based approach for defining CIs and their time-uniform variants called confidence sequences (CSs), has been shown to be empirically superior to the classical methods. In this paper, we provide theoretical justification for this improved empirical performance of betting CIs and CSs. Our main contributions are as follows: (i) We first compare CIs using the values of their first-order asymptotic widths (scaled by $\sqrt{n}$), and show that the betting CI of Waudby-Smith and Ramdas (2023) has a smaller limiting width than existing empirical Bernstein (EB)-CIs. (ii) Next, we establish two lower bounds that characterize the minimum width achievable by any method for constructing CIs/CSs in terms of certain inverse information projections. (iii) Finally, we show that the betting CI and CS match the fundamental limits, modulo an additive logarithmic term and a multiplicative constant. Overall these results imply that the betting CI~(and CS) admit stronger theoretical guarantees than the existing state-of-the-art EB-CI~(and CS); both in the asymptotic and finite-sample regimes.

翻译：通过独立同分布（i.i.d.）观测构建单变量分布均值的非渐近置信区间（CI）是统计学中的基本任务。对于有界观测，经典非参数方法通过反转标准浓度界（如霍夫丁不等式或伯恩斯坦不等式）来实现。近期，一种基于赌注的替代方法被提出用于定义CI及其时间均匀变体——置信序列（CS），并在实证中优于经典方法。本文为赌注CI和CS的改进实证性能提供了理论依据。主要贡献如下：（i）首先通过一阶渐近宽度（按$\sqrt{n}$缩放）比较CI，证明Waudby-Smith和Ramdas（2023）的赌注CI的极限宽度小于现有经验伯恩斯坦（EB）-CI。（ii）其次，建立两个下界，以基于特定逆信息投影刻画任何CI/CS构造方法所能达到的最小宽度。（iii）最后，证明赌注CI和CS匹配这些基本极限（模加性对数项和乘性常数）。总体而言，这些结果表明赌注CI（及CS）在渐近和有限样本场景下均比现有最优EB-CI（及CS）具有更强的理论保证。