On Conditional Mean Squared Errors of Empirical Bayes Estimators in Mixed Models with Application to Small Area Estimation

This paper is concerned with the prediction of the conditional mean which involves the fixed and random effects based on the natural exponential family with a quadratic variance function. The best predictor is interpreted as the Bayes estimator in the Bayesian context, and the empirical Bayes estimator (EB) is useful for small area estimation in the sense of increasing precision of prediction for small area means. When data of the small area of interest are observed and one wants to know the prediction error of the EB based on the data, the conditional mean squared error (cMSE) given the data is used instead of the conventional unconditional MSE. The difference between the two kinds of MSEs is small and appears in the second-order terms in the classical normal theory mixed model. However, it is shown that the difference appears in the first-order or leading terms for distributions far from normality. Especially, the leading term in the cMSE is a quadratic concave function of the direct estimate in the small area for the binomial-beta mixed model, and an increasing function for the the Poisson-gamma mixed model, while the leading terms in the unconditional MSEs are constants for the two mixed models. Second-order unbiased estimators of the cMSE are provided in two ways based on the analytical and parametric bootstrap methods. Finally, the performances of the EB and the estimator of cMSE are examined through simulation and empirical studies.