ROBUST TYPE GAUSS-MARKOV THEOREM AND RAO0S FIRST ORDER EFFICIENCY FOR THE SYMMETRIC TRIMMED MEAN

Abstract

In Chen and Chiang [2] and Chen, Thompson and Hung [3], the symmetric trimmed mean has been shown, for various linear models, to have the efficiency of having asymptotic covariance matrices close to the Cr\'amer-Rao lower bounds for some heavy tail error distributions. In this paper, we investigate some further theoretical results for this symmetric trimmed mean for the linear regression model. From the nonparametric point of view, we develop a robust version of the Gauss-Markov theorem for the problem of estimating regression parameter vector $\beta$ and parametric vector function $C\beta$ where the best estimators are this trimmed mean and $C$ multiplied by it, respectively. In addition, we show that these best estimators are the best Mallows-type bounded influence linear symmetric trimmed means. Finally, from the parametric aspect, we show that the symmetric trimmed mean is Rao$'$s first order efficient for a heavy tail error distribution.