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December, 1990 The Asymptotics of $S$-Estimators in the Linear Regression Model
Laurie Davies
Ann. Statist. 18(4): 1651-1675 (December, 1990). DOI: 10.1214/aos/1176347871


We consider the consistency and weak convergence of $S$-estimators in the linear regression model. Sufficient conditions for consistency with varying dimension are given which are sufficiently weak to cover, for example, polynomial trends and i.i.d. carriers. A weak convergence theorem for the Hampel-Rousseeuw least median of squares estimator is obtained, and it is shown under rather general conditions that the correct norming factor is $n^{1/3}$. Finally, the asymptotic normality of $S$-estimators with a smooth $\rho$-function is obtained again under weak conditions on the carriers.


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Laurie Davies. "The Asymptotics of $S$-Estimators in the Linear Regression Model." Ann. Statist. 18 (4) 1651 - 1675, December, 1990.


Published: December, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0719.62042
MathSciNet: MR1074428
Digital Object Identifier: 10.1214/aos/1176347871

Primary: 62J05
Secondary: 62F12 , 62F35

Keywords: $S$-estimators , asymptotic normality , consistency , least median of squares , Linear regression , weak convergence

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 4 • December, 1990
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