The Edgeworth expansion for U-statistics of degree two. (English) Zbl 0614.62015

Bickel, P. J.; Götze, F.; van Zwet, W. R.

Ann. Stat. 14, 1463-1484 (1986).

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 H. Callaert, P. Janssen and 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 \(o(N^{-1})\) for Wilcoxon’s one sample rank statistic is established under the assumptions that the common df F of the \(X_ i's\) is continuous and that the distribution of \(F(-X_ 1)\) has an absolutely continuous component.

Reviewer: R.Helmers

MSC:

62E20	Asymptotic distribution theory in statistics
60F05	Central limit and other weak theorems
62G99	Nonparametric inference

Keywords:

second order asymptotics; Edgeworth expansion; U-statistic with a kernel h of degree 2; eigenvalue condition; characteristic functions; conditioning argument; bound; moment inequality; concentration inequality; Wilcoxon’s one sample rank statistic

Citations:

Zbl 0428.62016