## The Annals of Statistics

### 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.

#### Article information

Source
Ann. Statist., Volume 9, Number 1 (1981), 194-200.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176345347

Digital Object Identifier
doi:10.1214/aos/1176345347

Mathematical Reviews number (MathSciNet)
MR600547

Zentralblatt MATH identifier
0457.62018

JSTOR
Callaert, Herman; Veraverbeke, Noel. The Order of the Normal Approximation for a Studentized $U$-Statistic. Ann. Statist. 9 (1981), no. 1, 194--200. doi:10.1214/aos/1176345347. https://projecteuclid.org/euclid.aos/1176345347