Preliminary-Test and Bayes Estimation of a Location Parameter Under 'Reflected Normal' Loss

In this paper, we consider a simple preliminary-test estimation problem where the analyst's loss structure is represented by a ‘reflected Normal' penalty function. In particular we consider the estimation of the location parameter in a Normal sampling problem, where a preliminary test is conducted for the validity of a simple restriction on this parameter. The exact finite-sample risk of this pre-test estimator is derived under ‘reflected Normal' loss, and this risk is compared with those of the unrestricted and restricted Maximum Likelihood estimators of location under this loss structure. The paper draws comparisons between these results and those obtained under conventional quadratic loss. Some simple Bayesian analysis is also considered. The results extend naturally to the case of estimating the coefficients in a Normal linear multiple regression model.