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：从独立同分布观测中构造单变量分布均值的非渐近置信区间是统计学中的基本任务。对于有界观测，经典的非参数方法通过反演标准浓度界限（如霍夫丁不等式或伯恩斯坦不等式）来实现。近期，一种基于博彩的替代方法（用于定义置信区间及其时间均匀变体——置信序列）已被证明在经验性能上优于经典方法。本文为博彩置信区间和置信序列的改进经验性能提供理论依据。主要贡献如下：（i）首先，我们通过一阶渐近宽度（以$\sqrt{n}$缩放）的值比较置信区间，并证明Waudby-Smith与Ramdas（2023）的博彩置信区间比现有经验伯恩斯坦置信区间具有更小的极限宽度；（ii）其次，我们建立两个下界，以某些逆信息投影的形式刻画任何构建置信区间/置信序列方法所能达到的最小宽度；（iii）最后，我们证明博彩置信区间和置信序列（除附加对数项和乘法常数外）匹配这些基本极限。总体而言，这些结果表明博彩置信区间（及置信序列）在渐近和有限样本框架下均具有比现有最优经验伯恩斯坦置信区间（及置信序列）更强的理论保证。