Abstract
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.
Citation
R. Helmers. "On the Edgeworth Expansion and the Bootstrap Approximation for a Studentized $U$-Statistic." Ann. Statist. 19 (1) 470 - 484, March, 1991. https://doi.org/10.1214/aos/1176347994
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