Optimum Robust Estimation of Linear Aspects in Conditionally Contaminated Linear Models

Abstract

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.