Statistica Sinica 11(2001), 147-172

ESTIMATORS FOR THE LINEAR REGRESSION MODEL

BASED ON WINSORIZED OBSERVATIONS

L-A Chen, A. H. Welsh and W. Chan

National Chiao Tung University, Australian National University

and University of Texas-Houston

Abstract: We develop an asymptotic, robust version of the Gauss-Markov theorem for estimating the regression parameter vector $\beta$ and a parametric function $c'\beta$ in the linear regression model. In a class of estimators for estimating $\beta$ that are linear in a Winsorized observation vector introduced by Welsh (1987), we show that Welsh$'$s trimmed mean has smallest asymptotic covariance matrix. Also, for estimating a parametric function $c'\beta$, the inner product of $c$ and the trimmed mean has the smallest asymptotic variance among a class of estimators linear in the Winsorized observation vector. A generalization of the linear Winsorized mean to the multivariate context is also given. Examples analyzing American lobster data and the mineral content of bones are used to compare the robustness of some trimmed mean methods.

Key words and phrases: Linear regression, robust estimation, trimmed mean, Winsorized mean.