Bayesian Point Estimation and Predictive Density Estimation for the Binomial Distribution with a Restricted Probability Parameter

TL;DR

This paper investigates Bayesian point and predictive density estimation for binomial distributions with restricted probability parameters, comparing different priors and establishing dominance conditions, supported by numerical studies.

Contribution

It introduces dominance conditions for Bayesian estimators under parameter restrictions and explores related Poisson problems, with comprehensive numerical analysis.

Findings

Dominance conditions for truncated vs. untruncated beta priors.

Extension to cases with both lower and upper bounds.

Numerical validation of theoretical results.

Abstract

In this paper, we consider Bayesian point estimation and predictive density estimation in the binomial case. After presenting preliminary results on these problems, we compare the risk functions of the Bayes estimators based on the truncated and untruncated beta priors and obtain dominance conditions when the probability parameter is less than or equal to a known constant. The case where there are both a lower bound restriction and an upper bound restriction is also treated. Then our problems are shown to be related to similar problems in the Poisson case. Finally, numerical studies are presented.