The Order of the Normal Approximation for a Studentized $U$-Statistic

Abstract

Let $U_N$ be a one sample $U$-statistic with kernel $h$ of degree two, such that $Eh(X_1, X_2) = \vartheta$ and $\operatorname{Var} E\lbrack h(X_1, X_2)\mid X_1 \rbrack > 0$. It is shown that for a studentized $U$-statistic $\sup_x|P(N^{1/2}S^{-1}_N(U_N - \vartheta) \leqslant x) - \Phi(x)| = O(N^{-1/2})$ as $N \rightarrow \infty$, where $N^{-1}S^2_N = 4N^{-1}(N - 1)(N - 2)^{-2}\sum^N_{i=1} \lbrack (N - 1)^{-1} \sum_{j\neq i} h(X_i, X_j) - U_N \rbrack^2$ is the jackknife estimator of $\operatorname{Var} U_N$. The condition needed to obtain this order bound is the existence of the 4.5th absolute moment of the kernel $h$. As in Helmers' Ph.D. thesis on linear combinations of order statistics, the analogous result for a studentized sum of i.i.d. random variables arises as a special case.