On conditional prediction errors in mixed models with application to small area estimation

The empirical Bayes estimators in mixed models are useful for small area estimation in the sense of increasing precision of prediction for small area means, and one wants to know the prediction errors of the empirical Bayes estimators based on the data. This paper is concerned with conditional prediction errors in the mixed models instead of conventional unconditional prediction errors. In the mixed models based on natural exponential families with quadratic variance functions, it is shown that the difference between the conditional and unconditional prediction errors is significant under distributions far from normality. Especially for the binomial–beta mixed and the Poisson–gamma mixed models, the leading terms in the conditional prediction errors are, respectively, a quadratic concave function and an increasing function of the direct estimate in the small area, while the corresponding leading terms in the unconditional prediction errors are constants. Second-order unbiased estimators of the conditional prediction errors are also derived and their performances are examined through simulation and empirical studies.