Bayesian and Non-Bayesian Estimation Using Balanced Loss Functions

Abstract

Balanced Loss Functions (BLFs) which reflect both goodness of fit and precision of estimation are described. Optimal estimates relative to BLFs are derived for estimation of a scalar mean, a vector mean and a vector of regression coefficients. Properties of these estimates are compared with those of maximum likelihood (ML) estimates and Bayesian posterior means. Sampling properties, or average operating characteristics, of optimal BLF estimates, ML estimates and posterior means are compared. The robustness of these estimates’ and estimators’ performance with respect to departures from the loss functions for which they are optimal is analyzed. It is concluded that BLFs and their associated optimal estimates will probably be useful in many estimation problems.