Predictive measures of the conflict between frequentist and Bayesian estimators

In the presence of prior information on an unknown parameter of a statistical model, Bayesian and frequentist estimates based on the same observed data do not coincide. However, in many standard parametric problems, this difference tends to decrease for growing sample size. In this paper we consider as a measure of discrepancy (Dn) the squared difference between Bayesian and frequentist point estimators of the parameter of a model. We derive the predictive distribution of Dn for finite sample sizes in the case of a one-dimensional exponential family and we study its behavior for increasing sample size. Numerical examples are illustrated for normal models.