Asymptotic normality of U-statistics based on trimmed samples
type: Item
instance of: scholarly article
title: Asymptotic normality of U-statistics based on trimmed samples
zbMATH Open document ID: 0617.62016
DOI: 10.1016/0378-3758(87)90056-5
author: Paul Janssen
author: Noël Veraverbeke
author: Robert J. Serfling
published in: Journal of Statistical Planning and Inference
publication date: 1987-01-01T00:00:00Z
review text: The authors consider asymptotic normality of U-statistics based on the trimmed sample. More precisely, let \(X_ 1,...,X_ n\) be an i.i.d. sample from a df F, let \(X_{n1}\leq...\leq X_{nn}\) denote the ordered \(X_ i's\), let \(0<\alpha\), \(\beta <1/2\), and put \(n_{\alpha,\beta}=n- [\alpha n]-[\beta n]\). \(Put\) U\({}_{n\alpha \beta}=\left( \begin{matrix} n_{\alpha \beta}\\ m\end{matrix} \right)^{-1}\sum_{C_{n\alpha \beta}}h(X_{ni_ 1},...,X_{ni_ m})\) where \(h(x_ 1,...,x_ m)\) is a kernel which is asymmetric in its arguments and \(C_{n\alpha \beta}\) denotes the set of m-tuples \(\{(i_ 1,...,i_ m):\) \([\alpha n]+1\leq i_ 1<...<i_ m\leq n-[\beta n])\). It is shown that under some regularity conditions \(U_{n\alpha \beta}\) is asymptotically normal. The result is extended to the multi-sample case. The authors' results unify the flexibility of the class of U-statistics and the theory of statistics based on trimmed samples. Various examples, such as moment-type statistics and Wilcoxon-type statistics, are also shown.
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62F35
Mathematics Subject Classification ID: 62G05
zbMATH DE Number: 4001183
zbMATH Keywords: asymmetric kernel
zbMATH Keywords: robust estimation
zbMATH Keywords: asymptotic normality
zbMATH Keywords: U-statistics
zbMATH Keywords: trimmed sample
zbMATH Keywords: multi-sample case
reviewed by: Ken-ichi Yoshihara
MaRDI profile type: Publication
full work available at URL: 0378-3758(87)90056-5
OpenAlex ID: W2004301828
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description: scientific article; zbMATH DE number 4001183