Priors for second-order unbiased Bayes estimators

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 several examples, including the linear regression model and the nested error regression (NER) 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 NER model, highlighting its effectiveness in small-sample settings.