A Bayesian approach to joint small area estimation.

Abstract

In small area estimation problems focus has been put on how to borrow strength across areas in order to develop a reliable estimator when auxiliary information is in hand. Some traditional methods for small area problems borrow strength through linear models that provide links to related areas, which may not be appropriate for some survey data. We propose a new approach to small area estimation, which borrows strength through a noninformative Bayesian prior without any assumption of linearity between variables. This approach results in a generalized constrained Dirichlet posterior estimator when auxiliary information is available for small areas. It is not only able to utilize the auxiliary information within small areas but also able to utilize the auxiliary information across small areas, which is usually impossible to take into account by traditional methods. When information about auxiliary variables is present, the proposed approach allows either estimates for a given area or, simultaneously, for several areas depending on the form of auxiliary information. The Bayes like character of the posterior allows one to prove the admissibility of the point estimator of interest suggesting that inferential procedures based on our approach will tend to have good frequentist properties. The form of our prior distribution allows us to assign a weight to each member of the sample and these weights allow us to find interval estimates for the small area means. This makes our methods easy to use in practice. Simulation studies and an application to a real study are given in this thesis to examine the performance of various approaches.