A Unified Measure of Uncertainty of Estimated Best Linear Unbiased Predictors in Small Area Estimation Problems

We obtain a second order approximation to the mean squared error (MSE), and its estimate, of the empirical or estimated best linear unbiased predictor (EBLUP) of a mixed effect in a general mixed linear normal model. This covers many important small area models in the literature. Unlike previous research in this area, we provide a unified theory of measuring uncertainty of an EBLUP for a complex small area model where the variance components are estimated by various standard methods including restricted or residual maximum likelihood (REML) and maximum likelihood (ML). It turns out that the MSE approximations for the REML and the ML methods are exactly the same in the second order asymptotic sense. However, the second order accurate estimator of this MSE based on the former method requires less bias correction than the one based on the latter method. This is due to a result in the paper which shows that the bias of the REML estimators of variance components is of lower order than that of the ML estimators. A simulation is undertaken to compare different methods of estimating the variance components and to study the properties of various estimators of the MSE of the mixed effect. In our context it is interesting to note that the residual likelihood is same as the conditional profile likelihood (CPL) of Cox and Reid (1987). Thus, this paper addresses an important open problem raised by Cox and Reid (1987) in small area prediction using the CPL method.