The Bayesian Posterior Estimators under Six Loss Functions for Unrestricted and Restricted Parameter Spaces

In this chapter, we have investigated six loss functions. In particular, the squared error loss function and the weighted squared error loss function that penalize overestimation and underestimation equally are recommended for the unrestricted parameter space (cid:1) ∞ ; ∞ ð Þ ; Stein ’ s loss function and the power-power loss function, which penalize gross overestimation and gross underestimation equally, are recommended for the positive restricted parameter space 0 ; ∞ ð Þ ; the power-log loss function and Zhang ’ s loss function, which penalize gross overestimation and gross underestimation equally, are recommended for 0 ; 1 ð Þ . Among the six Bayesian estimators that minimize the corresponding posterior expected losses (PELs), there exist three strings of inequalities. However, a string of inequalities among the six smallest PELs does not exist. Moreover, we summarize three hierarchical models where the unknown parameter of interest belongs to 0 ; ∞ ð Þ , that is, the hierarchical normal and inverse gamma model, the hierarchical Poisson and gamma model, and the hierarchical normal and normal-inverse-gamma model. In addition, we summarize two hierarchical models where the unknown parameter of interest belongs to 0 ; 1 ð Þ , that is, the beta-binomial model and the beta-negative binomial model. For empirical Bayesian analysis of the unknown parameter of interest of the hierarchical models, we use two common methods to obtain the estimators of the hyperparameters, that is, the moment method and the maximum likelihood estimator (MLE) method.