## The Annals of Statistics

- Ann. Statist.
- Volume 19, Number 1 (1991), 470-484.

### On the Edgeworth Expansion and the Bootstrap Approximation for a Studentized $U$-Statistic

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 19, Number 1 (1991), 470-484.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176347994

**Digital Object Identifier**

doi:10.1214/aos/1176347994

**Mathematical Reviews number (MathSciNet)**

MR1091863

**Zentralblatt MATH identifier**

0734.62049

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 62G05: Estimation 60F05: Central limit and other weak theorems

**Keywords**

Edgeworth expansions bootstrap approximations studentized $U$-statistics bootstrap confidence intervals Edgeworth based confidence intervals

#### Citation

Helmers, R. On the Edgeworth Expansion and the Bootstrap Approximation for a Studentized $U$-Statistic. Ann. Statist. 19 (1991), no. 1, 470--484. doi:10.1214/aos/1176347994. https://projecteuclid.org/euclid.aos/1176347994