The empirical best linear unbiased predictor (EBLUP) or the empirical Bayes estimator (EB) in the linear mixed model is recognized useful for the small area estimation, because it can increase the estimation precision by using the information from the related areas. Two of the measures of uncertainty of EBLUP is the estimation of the mean squared error (MSE) and the confidence interval, which have been studied under the second-order accuracy in the literature. This paper provides the general analytical results for these two measures in the unified framework, namely, we derive the conditions on the general consistent estimators of the variance components to satisfy the third-order accuracy in the MSE estimation and the 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 the other estimators including the Prasad-Rao and Fay-Herriot estimators in specific models.