Asymptotic optimality theory of confidence intervals of the mean

We address the classical problem of constructing confidence intervals (CIs) for the mean of a distribution, given \(N\) i.i.d. samples, such that the CI contains the true mean with probability at least \(1 - \delta\), where \(\delta \in (0,1)\). We characterize three distinct learning regimes based on the minimum achievable limiting width of any CI as the sample size \(N_{\delta} \to \infty\) and \(\delta \to 0\). In the first regime, where \(N_{\delta}\) grows slower than \(\log(1/\delta)\), the limiting width of any CI equals the width of the distribution's support, precluding meaningful inference. In the second regime, where \(N_{\delta}\) scales as \(\log(1/\delta)\), we precisely characterize the minimum limiting width, which depends on the scaling constant. In the third regime, where \(N_{\delta}\) grows faster than \(\log(1/\delta)\), complete learning is achievable, and the limiting width of the CI collapses to zero, converging to the true mean. We demonstrate that CIs derived from concentration inequalities based on Kullback--Leibler (KL) divergences achieve asymptotically optimal performance, attaining the minimum limiting width in both sufficient and complete learning regimes for distributions in two families: single-parameter exponential and bounded support. Additionally, these results extend to one-sided CIs, with the width notion adjusted appropriately. Finally, we generalize our findings to settings with random per-sample costs, motivated by practical applications such as stochastic simulators and cloud service selection. Instead of a fixed sample size, we consider a cost budget \(C_{\delta}\), identifying analogous learning regimes and characterizing the optimal CI construction policy.