Fourier and Hermite series estimates of regression functions
type: Item
instance of: scholarly article
title: Fourier and Hermite series estimates of regression functions
zbMATH Open document ID: 0623.62029
DOI: 10.1007/BF02481112
author: Włodzimierz Greblicki
author: M. Pawlak
published in: Annals of the Institute of Statistical Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let (X,Y) be a pair of random variables; X has a density f. A regression function \(m(x)=E\{Y| X=x\}\) is first estimated by \[ \hat m(x)=\sum^{n}_{j=1}Y_ jD_ N(x-X_ j)/\sum^{n}_{j=1}D_ N(x- X_ j), \] where \(\pi D_ N\) is the Dirichlet kernel of N-th oder, \((X_ j,Y_ j)\) are observations of (X,Y), N depends on n (a number of observations). The authors examine two cases of convergence \(\hat m\to m:\) ''in probability'' and ''almost completely''. There are given general conditions for these types of convergence. Formulas for the convergence rates are also given. The second type of estimation is analogous. \(D_ N\) is replaced here by \(d_ N(x,X_ j)\), where \(d_ N(x,y)\) is defined using Hermite polynomials. The modified convergence conditions give analogous convergence results as above.
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62J02
zbMATH DE Number: 4011659
zbMATH Keywords: regression function estimation
zbMATH Keywords: Fourier series
zbMATH Keywords: almost sure convergence
zbMATH Keywords: convergence in probability
zbMATH Keywords: Dirichlet kernel
zbMATH Keywords: convergence rates
zbMATH Keywords: Hermite polynomials
Wikidata QID: Q118592639
reviewed by: Pavel Froněk
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02481112
OpenAlex ID: W1965247739
description: scientific article; zbMATH DE number 4011659