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December, 1984 Asymptotic Behavior of M-Estimators of p Regression Parameters when p2/n is Large. I. Consistency
Stephen Portnoy
Ann. Statist. 12(4): 1298-1309 (December, 1984). DOI: 10.1214/aos/1176346793

Abstract

Consider the general linear model Y=xβ+R with Y and Rn-dimensional, βp-dimensional, and X an n×p matrix with rows xi. Let ψ be given and let ˆβ be an M-estimator of β satisfying 0=xiψ(Yixiˆβ). Previous authors have considered consistency and asymptotic normality of ˆβ when p is permitted to grow, but they have required at least p2/n0. Here the following result is presented: in typical regression cases, under reasonable conditions if p(logp)/n0 then ˆββ2=Op(p/n). A subsequent paper will show that ˆβ has a normal approximation in Rp if (plogp)3/2/n0 and that maxi|xi(ˆββ)|p0 (which would not follow from norm consistency if p2/n). In ANOVA cases, ˆβ is not norm consistent, but it is shown here that max|xi(ˆββ)|p0 if plogp/n0. A normality result for arbitrary linear combinations a(ˆββ) is also presented in this case.

Citation

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Stephen Portnoy. "Asymptotic Behavior of M-Estimators of p Regression Parameters when p2/n is Large. I. Consistency." Ann. Statist. 12 (4) 1298 - 1309, December, 1984. https://doi.org/10.1214/aos/1176346793

Information

Published: December, 1984
First available in Project Euclid: 12 April 2007

zbMATH: 0584.62050
MathSciNet: MR760690
Digital Object Identifier: 10.1214/aos/1176346793

Subjects:
Primary: 62G35
Secondary: 62E20 , 62J05

Keywords: M-estimators , asymptotic normality , consistency , General linear model , regression , robustness

Rights: Copyright © 1984 Institute of Mathematical Statistics

Vol.12 • No. 4 • December, 1984
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