A comparison of Adjusted Bayes Estimators of an Ensemble of Small Area Parameters

Bayesian estimators of small area parameters may be very effective in improving the precision of “direct”, design-based estimates. By the way, the may be poor estimators of the actual Empirical Distribution Function of the ensemble of small area parameters. In this paper we review different adjusted estimators that correct or mitigate this problem in the context of Hierarchical Bayesian modelling, considering also extensions to the case of multivariate models that are widely used in Small Area estimation. In particular we consider constrained and linear constrained Hierarchical Bayes estimators, along with an estimation method recently by Zhang. The posterior Mean Square Errors are proposed as measures of uncertainty. Different methods are compared by means of a simulation exercise in which many situations occurring in small area estimation applied problems are artificially re-created. The main goals of the simulation is to assess the ability of various estimators to properly estimate the actual Empirical Distribution Function of area parameters and secondly to assess the frequentist properties of posterior MSEs. The method of Zhang emerges as best in both the univariate and the multivariate setting, although the picture in the latter case is less clear.