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March, 1992 Optimum Robust Estimation of Linear Aspects in Conditionally Contaminated Linear Models
Viktor Kurotschka, Christine Muller
Ann. Statist. 20(1): 331-350 (March, 1992). DOI: 10.1214/aos/1176348525


P. J. Bickel's approach to and results on estimating the parameter vector $\beta$ of a conditionally contaminated linear regression model by asymptotically linear (AL) estimators $\hat\beta^\ast$ which have minimum trace of the asymptotic covariance matrix among all AL estimators with a given bound $b$ on their asymptotic bias (MT-AL estimators with bias bound $b$) is here extended to conditionally contaminated general linear models and in particular for estimating arbitrary linear aspects $\varphi(\beta) = C\beta$ of $\beta$ which are of actual interest in applications. Admitting that $\beta$ itself is not identifiable in the model (also a practically important situation), a complete characterization of MT-AL estimators with bias bound $b$ including the case where $b$ is smallest possible is presented here, which extends and sharpens H. Rieder's characterization of all AL estimators with minimum asymptotic bias. These characterizations (Theorem 1) represent generalizations (in different directions) of those which define Hampel-Krasker estimators for $\beta$ in linear regression models and admit (Theorem 2) explicit constructions of MT-AL estimators under generally applicable model assumption. Obviously, even in linear regression models, $\hat\varphi^\ast = C\hat\beta^\ast$ is not an MT-AL estimator for $\varphi$ if $\hat\beta^\ast$ is one for $\beta$ (there does not even exist an AL estimator nor an $M$ estimator for $\beta$, if $\beta$ is not identifiable in the model). Examples such as quadratic regression illustrate the not at all obvious relation between $\hat\beta^\ast$ and $\hat\varphi^\ast$, demonstrate the applicability of the general results and show explicitly the influence of the parametrization and the underlying design of the linear model.


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Viktor Kurotschka. Christine Muller. "Optimum Robust Estimation of Linear Aspects in Conditionally Contaminated Linear Models." Ann. Statist. 20 (1) 331 - 350, March, 1992.


Published: March, 1992
First available in Project Euclid: 12 April 2007

zbMATH: 0792.62024
MathSciNet: MR1150347
Digital Object Identifier: 10.1214/aos/1176348525

Primary: 62F35
Secondary: 62G05 , 62J05

Keywords: conditional contamination , Hampel-Krasker estimator , linear aspect , linear model , quadratic regression , robust estimation

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 1 • March, 1992
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