关键词: 广义最小二乘估计, Gauss-Markov定理, 稳健性, 同变估计, 对称凸函数, 广义最小二乘估计, Gauss-Markov定理, 稳健性, 同变估计, 对称凸函数

Abstract: Linear regression model with elliptically symmetric errors and unknown dispersion matrix was discussed. For a given matrix $ \Sigma}_{0}$, when the real dispersion matrix varying within certain range, the GLSE $\hat{\beta}({\vec \Sigma}_{0}) = (\X'{\vec \Sigma}_{0}^{-1}\X)^{-1}\X'{\vec \Sigma}_{0}^{-1}y$ is the minimum risk estimator under a large class of loss functions, which implies the GLSE is a robust estimator with respect to dispersion matrix and loss functions.

Key words: Gauss-Markov theorem, robust, equivariant estimator, symmetric convex function, generalized least squares estimator, Gauss-Markov theorem, robust, equivariant estimator, symmetric convex function