Bayesian Estimators for Small Area Models Shrinking Both Means and Variances

TL;DR

This paper introduces a Bayesian method for small area estimation that jointly shrinks both means and variances, addressing the realistic scenario where sampling variances are estimated rather than known, and demonstrates its effectiveness through simulations and empirical data.

Contribution

It proposes an objective Bayesian approach with a hierarchical model for sampling variances in the Fay-Herriot model, ensuring proper posterior inference.

Findings

The method produces accurate shrinkage estimates for means and variances.

Simulation studies show improved estimation accuracy over traditional methods.

Empirical analysis confirms practical applicability and robustness.

Abstract

For small area estimation of area-level data, the Fay-Herriot model is extensively used as a model based method. In the Fay-Herriot model, it is conventionally assumed that the sampling variances are known whereas estimators of sampling variances are used in practice. Thus, the settings of knowing sampling variances are unrealistic and several methods are proposed to overcome this problem. In this paper, we assume the situation where the direct estimators of the sampling variances are available as well as the sample means. Using these information, we propose a Bayesian yet objective method producing shrinkage estimation of both means and variances in the Fay-Herriot model. We consider the hierarchical structure for the sampling variances and we set uniform prior on model parameters to keep objectivity of the proposed model. For validity of the posterior inference, we show under mild conditions that the posterior distribution is proper and has finite variances. We investigate the numerical performance through simulation and empirical studies.