On dispersion preserving estimation of the mean of a binary variable from small areas

Over-shrinkage is a common problem in small area (or domain) estimation. It happens when the estimated small-area parameters have less between-area variation than their true values. To deal with this problem, Louis (1984), Ghosh (1992) and Spjøtvoll and Thomsen (1987) have proposed various constrained empirical and hierarchical Bayes methods. In this paper we study two non-Bayesian methods based on, respectively, the synthetic estimator and a variance-component model. We show first that the synthetic estimator entails loss of dispersion in general, from which it follows that the coverage level of the confidence intervals could be far below the nominal level of confidence, when these are derived from the sampling error alone. A bivariate variance-component model at the area-level, as well as its simplification, can greatly improve the efficiency of the confidence intervals. However, super-population approaches as such are unable to capture the distribution of the true area-parameters. We develop a finite-population approach based on an empirical finite-population distribution function of the area-parameters, which provides the necessary adjustment. The various methods will be illustrated using the data of the Census 1990. Finally, we notice that several European countries will base the upcoming Census on their administrative register systems, instead of collecting the information in the field. Improved small area estimation methods may prove to be valuable for assessing the quality of such Register Counting.