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

We first show that the Generalized Least Squares estimator is the best median unbiased estimator of the regression parameters for quite general loss functions, when the parameter space is unrestricted. Of note is the fact that this result holds without moment restrictions. Thus, the errors may have multivariate Cauchy distribution. Next, we show that a restricted GLS estimator is best median unbiased for a linear combination of the regression parameters, when that linear combination is restricted to lie in an interval. Certain other linear combinations of the parameter vector may be subject to arbitrary additional restrictions. The paper then presents best median unbiased estimators of the error variance sigma-squared, as well as monotone functions of sigma-squared, when the errors are normally distributed. If sigma-squared is constrained to lie in a finite interval, the best estimator is a censored version of its unconstrained counterpart. When sigma-square is constrained only to be positive, the best median unbiased estimator is always larger than the best mean unbiased estimator s-squared, and is approximately equal to s-squared calculated with its degrees of freedom reduced by .66.