On the Bayesian Estimation for the Uniform Scale Parameter via a Functional Equation

Abstract

Bayes uniform model under the squared error loss function is shown to be completely identifiable by the form of the Bayes estimates of the scale parameter. This results in solving a specific functional equation. A complete characterization of differentiable Bayes estimators (BE) and generalized Bayes estimators (GBE) is given as well as relations between degrees of smoothness of the estimators and the priors. Characterizations of strong (generalized Bayes) Bayes sequence (SBS or SGBS) are also investigated. A SBS is a sequence of estimators (one for each sample size) where all its components are BE generated by the same prior measure. A complete solution is given for polynomial Bayesian estimation.

Keywords:

• Bayesian analysis
• Bayes and generalized Bayes estimators
• Characterization theorems
• Scale parameters
• Strong Bayes sequences

Mathematics Subject Classification:

• Primary 62C10
• Secondary 62H05

Acknowledgments

The authors are grateful to the referees because their remarks helped to improve the presentation of the material. They acknowledge the financial support partially provided by project BEC2004-3303.