Constrained empirical Bayes estimator and its uncertainty in normal linear mixed models

The empirical Bayes (EB) estimator or empirical best linear unbiased predictor (EBLUP) in the linear mixed model (LMM) is useful for the small area estimation in the sense of increasing the precision of estimation of small area means. However, one potential difficulty of EB is that when aggregated, the overall estimate for a larger geographical area may be quite different from the corresponding direct estimate like the overall sample mean. One way to solve this problem is the benchmarking approach, and the constrained EB (CEB) is a feasible solution which satisfies the constraints that the aggregated mean and variance are identical to the requested values of mean and variance. An interesting query is whether CEB may have a larger estimation error than EB. In this paper, we address this issue by deriving asymptotic approximations of MSE of CEB. Also, we provide asymptotic unbiased estimators for MSE of CEB based on the parametric bootstrap method, and establish their second-order justification. Finally, the performance of the suggested MSE estimators is numerically investigated.