Necessary and sufficient consistency conditions for a recursive kernel regression estimate
type: Item
instance of: scholarly article
title: Necessary and sufficient consistency conditions for a recursive kernel regression estimate
zbMATH Open document ID: 0627.62040
DOI: 10.1016/0047-259X(87)90178-3
author: Włodzimierz Greblicki
author: M. Pawlak
published in: Journal of Multivariate Analysis
publication date: 1987-01-01T00:00:00Z
review text: A recursive kernel estimate \(\sum^{n}_{i=1}Y_ iK((x-X_ i)/h_ i)/\sum^{n}_{j=1}K((x-X_ j)/h_ j)\) of a regression \(m(x)=E\{Y| X=x\}\) calculated from independent observations \((X_ 1,Y_ 1),...,(X_ n,Y_ n)\) of a pair (X,Y) of random variables is examined. For \(E| Y| <\infty\), the estimate is weakly pointwise consistent for almost all (\(\mu)\) \(x\in R^ d\), \(\mu\) is the probability measure of X, if and only if \[ \sum^{n}_{i=1}h^ d_ iI_{\{h_ i>\epsilon \}}/\sum^{n}_{j=1}h^ d_ j\to 0\quad as\quad n\to \infty, \] all \(\epsilon >0\), and \(\sum^{\infty}_{i=1}h^ d_ i=\infty\), d is the dimension of X. For \(E| Y|^{1+\delta}<\infty\), \(\delta >0\), the estimate is strongly pointwise consistent for almost all (\(\mu)\) \(x\in R^ d\), if and only if the same conditions hold. For \(E| Y|^{1+\delta}<\infty\), \(\delta >0\), weak and strong consistency are equivalent. Similar results are given for complete convergence.
Mathematics Subject Classification ID: 62G05
zbMATH DE Number: 4020218
zbMATH Keywords: regression function
zbMATH Keywords: recursive kernel estimate
zbMATH Keywords: independent observations
zbMATH Keywords: weakly pointwise consistent
zbMATH Keywords: strongly pointwise consistent
zbMATH Keywords: consistency
zbMATH Keywords: complete convergence
MaRDI profile type: Publication
Wikidata QID: Q118592634
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full work available at URL: 0047-259x(87)90178-3
OpenAlex ID: W2070045967
description: scientific article; zbMATH DE number 4020218