Pre-test estimation in regression under absolute error loss

Abstract

We consider the risks of the Ordinary Least Squares, Restricted Least Squares and Pre-Test estimators of a regression coefficient under absolute error loss. These results are compared with their quadratic loss counterparts, and similar regions of risk dominance are found to hold, at least qualitatively. * This note is based on some ongoing collaborative research being undertaken in this general field with Offer Lieberman. I am most grateful to him for his substantial input, and to Judith Giles and Kazuhiro Ohtani for many helpful discussions.

Further results on optimal critical values of pre‐test when estimating the regression error variance

“…In recent years, many of the standard results in the pre‐test literature have been reconsidered in the context of different loss functions. Some examples using absolute error loss, LINEX loss and balanced loss include the contributions of Giles (1993), Giles and Giles (1993b), Wan (1994a), Ohtani et al (1997), among others 2 . In a recent article, Wan and Zou (2003) proved a general result which suggests that for certain pre‐testing problems, the same critical value may be optimal for all first‐order differentiable loss functions provided that a mild technical condition is satisfied.…”