Coverage errors for Student's t confidence intervals comparable to those in Hall (1988)

Table 1 of Hall (1988) contains asymptotic coverage error formulas for some nonparametric approximate 95\% confidence intervals for the mean based on $n$ IID samples. The table includes an entry for an interval based on the central limit theorem using Gaussian quantiles and the Gaussian maximum likelihood variance estimate. It is missing an entry for the very widely used Student's $t$ confidence intervals. This note develops such a formula. The impetus to revisit this issue arose from the surprisingly robust performance of confidence intervals based on Student's t statistic in randomized quasi-Monte Carlo sampling. Hall's table had $0.14\kappa -2.12\gamma^2-3.35$ for normal theory intervals; the corresponding entry for Student's $t$ is $0.14\kappa -2.12\gamma^2$. An earlier version of this note reported that it corrected some coverage error formulas in Hall (1988). Two-sided errors take the form $2\Phi^{-1}(0.975)(A\kappa + \gamma^2+C)\varphi(1.96)/n +O(1/n^{3/2})$ where the error may well be $O(n^{-2})$. Hall's table showed $\Phi^{-1}(0.975)(A\kappa + B\gamma^2+C)$. The version intended as a correction had $2(A\kappa + B\gamma^2+C)$, wider by about $2/1.96\doteq1.02$. So, Hall's table really is proportional to the two-sided coverage errors.