The bootstrap estimate of distribution functions of studentized statistics is shown to be more accurate than even the two-term empirical Edgeworth expansion, thus strengthening the claim of superiority of the bootstrap over the normal approximation method. The two methods are compared not only with respect to bounded bowl-shaped loss functions but also with respect to squared error loss and, more generally, in $L^p$-norms.
"Second Order and $L^p$-Comparisons between the Bootstrap and Empirical Edgeworth Expansion Methodologies." Ann. Statist. 17 (1) 160 - 169, March, 1989. https://doi.org/10.1214/aos/1176347008