Empirical Uncertain Bayes Methods in Area-level Models

TL;DR

This paper introduces an empirical uncertain Bayes (EUB) method for small area estimation that accounts for uncertain prior distributions, improving estimation accuracy across various models like Poisson-gamma, binomial-beta, and Fay-Herriot.

Contribution

It develops an EUB approach using mixture priors and EM algorithm for hyperparameter estimation, enhancing small area mean estimation accuracy.

Findings

EUB estimator outperforms traditional empirical Bayes in simulations.

Second-order unbiased MSE estimator effectively evaluates risk.

Real data applications demonstrate practical advantages of EUB.

Abstract

Random effects model can account for the lack of fitting a regression model and increase precision of estimating area-level means. However, in case that the synthetic mean provides accurate estimates, the prior distribution may inflate an estimation error. Thus it is desirable to consider the uncertain prior distribution, which is expressed as the mixture of a one-point distribution and a proper prior distribution. In this paper, we develop an empirical Bayes approach for estimating area-level means, using the uncertain prior distribution in the context of a natural exponential family, which we call the empirical uncertain Bayes (EUB) method. The regression model considered in this paper includes the Poisson-gamma and the binomial-beta, and the normal-normal (Fay-Herriot) model, which are typically used in small area estimation. We obtain the estimators of hyperparameters based on the marginal likelihood by using a well-known Expectation-Maximization algorithm, and propose the EUB estimators of area means. For risk evaluation of the EUB estimator, we derive a second-order unbiased estimator of a conditional mean squared error by using some techniques of numerical calculation. Through simulation studies and real data applications, we evaluate a performance of the EUB estimator and compare it with the usual empirical Bayes estimator.