Estimators for the linear regression model based on Winsorized observations

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.