The Edgeworth expansion for U-statistics of degree two
type: Item
instance of: scholarly article
title: The Edgeworth expansion for U-statistics of degree two
zbMATH Open document ID: 0614.62015
published in: The Annals of Statistics
publication date: 1986-01-01T00:00:00Z
review text: An Edgeworth expansion with remainder \(o(N^{-1})\) is established for a U-statistic with a kernel h of degree 2 under very mild assumptions that are easy to verify and do not involve smoothness of the distribution of \(h(X_ 1,X_ 2).\) Define \(g(x)=E(h(X_ 1,X_ 2)| X_ 1=x)\), \(\psi (x,y)=h(x,y)-g(x)- g(y)\) and let \(Eh(X_ 1,X_ 2)=0\). Suppose that there exist a number \(r>2\) and an integer k such that \((r-2)(k-4)>8\) and that the following assumptions are satisfied: \(E| \psi (X_ 1,X_ 2)|^ r<\infty\), \(E| g(X_ 1)|^ 4<\infty\), \(\limsup_{| t| \to \infty}| Ee^{itg(X_ 1)}| <1\), and \(\psi\) possesses k non zero eigenvalues with respect to \(P_ X\), the common distribution of \(X_ 1,X_ 2,... \). Under these conditions, the validity of the Edgeworth expansion with remainder \(o(N^{-1})\) for a U-statistic with a kernel h of degree 2 is established. It is noted that a useful sufficient condition implying the authors' eigenvalue condition is that there exist points \(y_ 1,...,y_ k\) in the support of the common distribution of \(X_ 1,X_ 2,...\), such that the functions \(\psi (.,y_ 1),...,\psi (.,y_ k)\) are linearly independent. The authors' proof employs characteristic functions (ch.f.) methods. The analysis of the ch.f. of a U-statistic is done separately for small (and intermediate) and for large values of the argument. For small values of the argument the treatment of the ch.f. is fairly standard in view of the earlier work by \textit{H. Callaert}, \textit{P. Janssen} and \textit{N. Veraverbeke} [ibid. 8, 299-312 (1980; Zbl 0428.62016)]. However, essential difficulties arise when dealing with the ch.f. of a U-statistic for large values of the argument. Here the authors employ a rather delicate conditioning argument to obtain a bound for the ch.f. of a U- statistic. Also a moment inequality and a concentration inequality, which may be of wider interest, are needed in the proof. As an application of the authors' main result, the Edgeworth expansion with remainder…
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 62G99
zbMATH DE Number: 3994733
zbMATH Keywords: second order asymptotics
zbMATH Keywords: Edgeworth expansion
zbMATH Keywords: U-statistic with a kernel h of degree 2
zbMATH Keywords: eigenvalue condition
zbMATH Keywords: characteristic functions
zbMATH Keywords: conditioning argument
zbMATH Keywords: bound
zbMATH Keywords: moment inequality
zbMATH Keywords: concentration inequality
zbMATH Keywords: Wilcoxon's one sample rank statistic
author: P. J. Bickel
author: Friedrich Götze
author: Willem van Zwet
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176350170
OpenAlex ID: W4256748210
DOI: 10.1214/AOS/1176350170
description: scientific article; zbMATH DE number 3994733