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.

基于赌注方法构建的置信区间和置信序列在理论上具有更强的保证，无论是在渐近还是有限样本情况下，其经验证明在经验性能上优于现有的经验伯恩斯坦置信区间和置信序列。