Best Median-Unbiased Estimation in Linear Regression with Bounded Asymmetric Loss Functions

Abstract

Abstract This article considers optimal median-unbiased estimation in a linear regression model with the distribution of the errors lying in a subclass of the elliptically symmetric distributions. The generalized least squares (GLS) estimator is shown to be best for any monotone loss function, that is, any loss function that is nondecreasing as the magnitude of underestimation or overestimation increases. This includes bounded asymmetric loss functions. For the same loss functions, a restricted GLS estimator is shown to be best when the estimand is known to lie in an interval. For the case of normal errors, a best median-unbiased estimator of the error variance σ2 is given for restricted and unrestricted parameter spaces. This estimator differs from the sample variance s 2. In comparison with best mean-unbiased estimators of regression and variance parameters, the best median-unbiased estimators considered here take advantage of restrictions on the parameter space and are optimal with respect to a much wi...