Benchmarked empirical Bayes methods in multiplicative area-level models with risk evaluation

The paper develops hierarchical empirical Bayes and benchmarked hierarchical empirical Bayes estimators of positive small area means under multiplicative models. The usual benchmarking requirement is that the small area estimates, when aggregated, should equal the direct estimates for the larger geographical areas. However, while estimating positive small area parameters, the conventional squared error or weighted squared error loss subject to the usual benchmark constraint may not produce positive estimators, so it is necessary to seek other loss functions. We consider a multiplicative model for the original data for estimating positive small area means, and suggest a variant of the Kullback–Leibler divergence as a loss function. The prediction errors of the suggested hierarchical empirical Bayes estimators are investigated asymptotically, and their second-order unbiased estimators are provided. Bootstrapped estimators of these prediction errors for both hierarchical empirical Bayes and benchmarked hierarchical empirical Bayes estimators are also given. The performance of the suggested procedures is investigated through simulation as well as with an example.