The mean squared error of small area predictors constructed with estimated area variances.

*(English)*Zbl 1046.62071Summary: In the small area estimation literature, the sampling error variances are customarily assumed to be known or to depend on a finite number of parameters. We consider the empirical best linear unbiased predictor (EBLUP) obtained by using the individual directly estimated variance for each small area. An approximation for the mean squared error (MSE) of the EBLUP that recognizes the impact on the predictors of estimation of the variance components is derived. Simulation studies show that the theoretical expressions are good approximations for the MSE of the predictors unless the between-area variance component is very small (relative to the within-area variance). An improved estimator of the MSE is developed that has smaller overestimation than the orielnal estimator when the between-area variance component is small. The robustness of the MSE estimator is studied and predictors for nonnormal sampling errors are proposed. An example from the National Resources Inventory that motivated the development of the theory is described.