On estimation with balanced loss functions

Zellner ((1994) in: Gupta, S.S., Berger, J.O. (Eds.), Statistical Decision Theory and Related Topics. Springer, New York, pp. 371-390), introduced the notion of a balanced loss function in the context of a general linear model to reflect both goodness of fit and precision of estimation. We study this notion from the perspective of unifying a variety of results both frequentist and Bayesian. We show in broad generality that frequentist and Bayesian results for balanced loss follow from and also imply related results for quadratic loss functions reflecting only precision of estimation. Several examples are given for normal error structures and more generally for spherically symmetric error distribution.