On the Edgeworth expansion and the bootstrap approximation for a studentized U-statistic

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(X1) = E[h(Xl, X2)1X11. As an application improved Edgeworth and bootstrap based confidence intervals for the mean of a U-statistic are obtained.

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