On Measuring Uncertainty of Small Area Estimators with Higher Order Accuracy

Abstract

The empirical best linear unbiased predictor (EBLUP) or the empirical Bayes estimator (EB) in the linear mixed model is recognized as being useful for small area estimation, because it can increase the estimation precision by using information from the related areas. Two of the measures of uncertainty of the EBLUP are the estimation of the mean squared error (MSE) and the confidence interval, which have been studied under second-order accuracy in the literature. This paper provides general and advanced analytical results for these two measures in the unified framework. Namely, we derive the conditions on the general consistent estimators of the variance components so as to establish third-order accuracy in the estimation of the MSE and confidence interval in the general linear mixed normal models. Those conditions are shown to be satisfied by not only the maximum likelihood (ML) and restricted maximum likelihood (REML), but also other estimators including the Prasad-Rao and Fay-Herriot estimators in specific models.