Asymptotic behavior of M-estimators of p regression parameters when \(p^ 2/n\) is large. I. Consistency
type: Item
instance of: scholarly article
title: Asymptotic behavior of M-estimators of p regression parameters when \(p^ 2/n\) is large. I. Consistency
zbMATH Open document ID: 0584.62050
published in: The Annals of Statistics
publication date: 1984-01-01T00:00:00Z
review text: M-estimation of the regression parameters in the general linear model \(Y_ i=\sum^{p}_{j=1}\beta_ jx_{ji}+R_ i\) is defined as the solution to the system of equations \[ \sum^{n}_{i=1}x_{ji}\psi (Y_ i-\sum^{p}_{j=1}\beta_ jx_{ji}),\quad j=1,...,p. \] This paper considers asymptotic properties of M-estimators, \({\hat \beta}\). In the case of linear regression it is shown that if \(\psi\) is increasing, p(log p)/n\(\to 0\), and some other relatively mild conditions hold, then \(\| {\hat \beta}\|^ 2=O_ p(p/n)\). In the analysis of variance case of the general linear model it is shown that if p(log p)/n\(\to 0\) then at least \(\max_ j| {\hat \beta}_ j| =O_ p((p(\log p)/n)^{1/2})\). Also a result giving asymptotic normality for arbitrary linear combinations a'\({\hat \beta}\) is presented.
reviewed by: Hans Nyquist
Mathematics Subject Classification ID: 62F35
Mathematics Subject Classification ID: 62J05
Mathematics Subject Classification ID: 62J10
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 3936209
zbMATH Keywords: consistency
zbMATH Keywords: robustness
zbMATH Keywords: M-estimation
zbMATH Keywords: general linear model
zbMATH Keywords: asymptotic normality
zbMATH Keywords: linear combinations
author: Stephen Portnoy
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176346793
OpenAlex ID: W2017413905
DOI: 10.1214/AOS/1176346793
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description: scientific article; zbMATH DE number 3936209