In this paper the validity of a one-term Edgeworth expansion for Studentized symmetric statistics is proved. We propose jackknife estimates for the unknown constants appearing in the expansion and prove their consistency. As a result we obtain the second-order correctness of the empirical Edgeworth expansion for a very general class of statistics, including $U$-statistics, $L$-statistics and smooth functions of the sample mean. We illustrate the application of the bootstrap in the case of a $U$-statistic of degree two.
"Empirical Edgeworth expansions for symmetric statistics." Ann. Statist. 26 (4) 1540 - 1569, August 1998. https://doi.org/10.1214/aos/1024691253