The asymptotic accuracy of the estimated one-term Edgeworth expansion and the bootstrap approximation for a Studentized $U$-statistic is investigated. It is shown that both the Edgeworth expansion estimate and the bootstrap approximation are asymptotically closer to the exact distribution of a Studentized $U$-statistic than the normal approximation. The conditions needed to obtain these results are weak moment assumptions on the kernel $h$ of the $U$-statistic and a nonlattice condition for the distribution of $g(X_1) = E\lbrack h(X_1, X_2) \mid X_1\rbrack$. As an application improved Edgeworth and bootstrap based confidence intervals for the mean of a $U$-statistic are obtained.
@article{1176347994,
author = {Helmers, R.},
title = {On the Edgeworth Expansion and the Bootstrap Approximation for a Studentized $U$-Statistic},
journal = {Ann. Statist.},
volume = {19},
number = {1},
year = {1991},
pages = { 470-484},
language = {en},
url = {http://dml.mathdoc.fr/item/1176347994}
}
Helmers, R. On the Edgeworth Expansion and the Bootstrap Approximation for a Studentized $U$-Statistic. Ann. Statist., Tome 19 (1991) no. 1, pp. 470-484. http://gdmltest.u-ga.fr/item/1176347994/