## The Annals of Statistics

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

R. Helmers

#### 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
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