Title: Loss Functions in Restricted Parameter Spaces and Their Bayesian Applications,
Abstract: A squared error loss remains the most commonly used loss function for
constructing a Bayes estimator of the parameter of interest. It, however, can
lead to sub-optimal solutions when a parameter is defined on a restricted
space. It can also be an inappropriate choice in the context when an
overestimation and/or underestimation results in severe consequences and a more
conservative estimator is preferred. We advocate a class of loss functions for
parameters defined on restricted spaces which infinitely penalize boundary
decisions like the squared error loss does on the real line. We also recall
several properties of loss functions such as symmetry, convexity and
invariance. We propose generalizations of the squared error loss function for
parameters defined on the positive real line and on an interval. We provide
explicit solutions for corresponding Bayes estimators and discuss multivariate
extensions. Three well-known Bayesian estimation problems are used to
demonstrate inferential benefits the novel Bayes estimators can provide in the
context of restricted estimation.