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(X₁, X₂) X₁. As an application improved Edgeworth and bootstrap based confidence intervals for the mean of a U-statistic are obtained.