"Benchmarked Empirical Bayes Methods in Multiplicative Area-level Models with Risk Evaluation"

　　 The paper develops empirical Bayes and benchmarked empirical Bayes estimators of positive small area means under multiplicative models. A simple example will be estimation of per capita income for small areas. It is now well-understood that small area estimation needs explicit, or at least implicit use of models. One potential difficulty with model-based estimators is that the overall estimator for a larger geographical area based on (weighted) sum of the model-based estimators is not necessarily identical to the corresponding direct estimator, such as the overall sample mean. One way to fix such a problem is the so-called benchmarking approach which modifies the model-based estimators to match the aggregate direct estimator. Benchmarked hierarchical and empirical Bayes estimators have proved to be particularly useful in this regard. However, while estimating positive small area parameters, the conventional squared error or weighted squared loss subject to the usual benchmark constraint does not necessarily produce positive estimators. Hence, it is necessary to seek other meaningful losses to alleviate this problem. In this paper, we consider the transformed Fay-Herriot model as a multiplicative model for estimating positive small area means, and suggest a weighted Kullback-Leibler divergence as a loss function. We have found out that the resulting Bayes estimator is the posterior mean and that the corresponding benchmarked Bayes and empirical Bayes estimators retain the positivity constraint. The prediction errors of the suggested empirical Bayes estimators are investigated asymptotically, and their second-order unbiased estimators are provided. In addition, bootstrapped estimators of these prediction errors are also provided. The performance of the suggested procedures is investigated through simulation as well as with an empirical study.