Small area estimation: A Bayesian perspective.

Abstract

For the last 25 years the special problems of deriving estimates for small areas or domains (subsets of the entire population) from sample surveys have received increasing attention within the survey sampling literature. Many attempts to derive such estimators have been either ad hoc approaches for specific problems, assuming specific models for the data, or attempts to apply large-sample sampling theory to problems of small samples. In this dissertation existing estimators will be described, including Bayesian techniques that have been tried, and then a Bayesian approach will be applied to the small area estimation problem. The first chapter of the dissertation organizes the different estimators, summarizing them, and showing where certain methods can be viewed as minor variations or generalizations of others. From this review a clearer understanding of the present techniques and their interrelationships is obtained, as is a path toward a generalization of existing estimators. The synthetic estimator has been proposed for many applications where the survey data are too sparse to produce direct, design-unbiased estimates with the desired level of accuracy. In Chapter 2 a Bayesian framework is presented in which the synthetic estimator is the Bayes estimate under a series of limiting conditions, and for which a general form of the synthetic estimator (GSE) is appropriate for other situations. Chapter 3 describes how three other small area estimators can be derived as Bayes estimators for special cases of the same general framework used in Chapter 2 to derive the GSE. The robustness of the GSE is examined in the final section of this chapter. Existing procedures for producing mean square errors for model-based small area estimators are reviewed in Chapter 4. A procedure for developing small area specific design-based mean square errors for model-based estimators is introduced. Chapter 5 of the dissertation presents an empirical comparison of estimators using data from the U.S. National Health Interview Survey. For two different variables estimates are computed for multiple forms of the GSE and for two other estimators, for each of the 50 states and the District of Columbia. Mean square errors are also produced using the procedure developed in Chapter 4. The impact on these estimators of using national versus subnational models is examined. Chapter 6 summarizes the major findings and contributions of the dissertation. Situations in which the procedures developed in Chapters 4 and 5 are likely to be useful are described, as are topics for related future research.