川久保 友超

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.

In linear mixed models, the conditional Akaike Information Criterion (cAIC) is a procedure for variable selection in light of the prediction of specific clusters or random effects. This is useful in problems involving prediction of random effects such as small area estimation, and much attention has been received since suggested by Vaida and Blanchard (2005). A weak point of cAIC is that it is derived as an unbiased estimator of conditional Akaike Information (cAI) in the overspecified case, namely in the case that candidate models include the true model. This results in larger biases in the underspecified case that the true model is not included in candidate models. In this paper, we derive the modified cAIC (McAlC) to cover both the underspecified and overspecified cases, and investigate properties of McAlC. It is numerically shown that McAIC has less biases and less prediction errors than cAIC. (C) 2014 Elsevier Inc. All rights reserved.