We derive upper bounds for the coverage error of confidence intervals for a population mean uniformly over large classes of populations and different types of confidence intervals. It is shown that the order of these bounds is achieved by the normal approximation method for constructing confidence intervals, uniformly over distributions with finite third moment, and, by an empirical Edgeworth correction of this approach, uniformly over smooth distributions with finite fourth moments. These results have straightforward extensions to higher orders of Edgeworth correction and higher orders of moments. Our upper bounds to coverage accuracy are based on Berry-Esseen theorems for Edgeworth expansions of the distribution of the Studentized mean.
"Uniform Coverage Bounds for Confidence Intervals and Berry-Esseen Theorems for Edgeworth Expansion." Ann. Statist. 23 (2) 363 - 375, April, 1995. https://doi.org/10.1214/aos/1176324525