Priors for second-order unbiased Bayes estimators

TL;DR

This paper extends the concept of asymptotically unbiased priors to non-i.i.d. models, providing a systematic way to construct such priors and demonstrating their effectiveness in small-sample linear and nested error regression models.

Contribution

It generalizes Hartigan's framework to non-i.i.d. models, deriving PDEs for asymptotically unbiased priors and offering a construction procedure.

Findings

The method successfully constructs asymptotically unbiased priors for complex models.

Simulation results show improved small-sample performance of the Bayes estimator.

The approach applies to linear and nested error regression models.

Abstract

Asymptotically unbiased priors, introduced by Hartigan (1965), are designed to achieve second-order unbiasedness of Bayes estimators. This paper extends Hartigan's framework to non-i.i.d. models by deriving a system of partial differential equations that characterizes asymptotically unbiased priors. Furthermore, we establish a necessary and sufficient condition for the existence of such priors and propose a simple procedure for constructing them. The proposed method is applied to the linear regression model and the nested error regression model (also known as the random effects model). Simulation studies evaluate the frequentist properties of the Bayes estimator under the asymptotically unbiased prior for the nested error regression model, highlighting its effectiveness in small-sample settings.