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Optimum robust estimation of linear aspects in conditionally contaminated linear models. (English) Zbl 0792.62024

Summary: P. J. Bickel’s approach [ibid. 12, 1349-1368 (1984; Zbl 0567.62051); Lect. Notes Math. 876, 2-72 (1981; Zbl 0484.62053)] to and results on estimating the parameter vector \(\beta\) of a conditionally contaminated linear regression model by asymptotically linear (AL) estimators \(\widehat{\beta}^*\) 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 [Stat. Decis. 5, 307-336 (1987; Zbl 0631.62035); Robust estimation of functionals. Tech. Rep., Univ. Bayreuth (1985)] characterization of AL estimators with minimum asymptotic bias. These characterizations represent generalizations (in different directions) of those which define Hampel-Krasker estimators [see F. R. Hampel et al., Robust statistics. The approach based on influence functions. (1986; Zbl 0593.62027); W. S. Krasker, Econometrica 48, 1333-1346 (1980; Zbl 0467.62096)] for \(\beta\) in linear regression models and admit explicit constructions of MT-AL estimators under generally applicable model assumptions. Obviously, even in linear regression models, \(\widehat{\varphi}^*= C\widehat{\beta}^*\) is not an MT-AL estimator for \(\varphi\) if \(\widehat{\beta}^*\) 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 \(\widehat{\beta}^*\) and \(\widehat{\varphi}^*\), demonstrate the applicability of the general results and show explicitly the influence of the parametrization and the underlying design of the linear model.

MSC:

62F35 Robustness and adaptive procedures (parametric inference)
62J05 Linear regression; mixed models
62F10 Point estimation
62K99 Design of statistical experiments
62G35 Nonparametric robustness
62G07 Density estimation
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