The Estimation of the Mean Squared Error of Small-Area Estimators

Abstract

Abstract Small-area estimation has received considerable attention in recent years because of a growing demand for reliable small-area statistics. The direct-survey estimators, based only on the data from a given small area (or small domain), are likely to yield unacceptably large standard errors because of small sample size in the domain. Therefore, alternative estimators that borrow strength from other related small areas have been proposed in the literature to improve the efficiency. These estimators use models, either implicitly or explicitly, that connect the small areas through supplementary (e.g., census and administrative) data. For example, simple synthetic estimators are based on implicit modeling. In this article, three small-area models, of Battese, Harter, and Fuller (1988), Dempster, Rubin, and Tsutakawa (1981), and Fay and Herriot (1979), are investigated. These models are all special cases of a general mixed linear model involving fixed and random effects, and a small-area mean can be expressed as a linear combination of fixed effects and realized values of random effects. Using the general theory of Henderson (1975) for a mixed linear model, a two-stage estimator (or predictor) of a small-area mean under each model is obtained, by first deriving the best linear unbiased estimator (or predictor) assuming that the variance components that determine the variance-covariance matrix are known, and then replacing the variance components in the estimator with their estimators. Second-order approximation to the mean squared error (MSE) of the two-stage estimator and the estimator of MSE approximation are obtained under normality. Finally, the results of a Monte Carlo study on the efficiency of two-stage estimators and the accuracy of the proposed approximation to MSE and its estimator are summarized. The MSE approximation provides a reliable measure of uncertainty associated with the two-stage estimator. It can also provide asymptotically valid confidence intervals on a small-area mean, as the number of small areas tends to ∞.