The Bayesian and Approximate Bayesian Methods in Small Area Estimation
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For small area estimation, model based methods are preferred to the traditional design based methods because of their ability to borrow strength from related sources. The indirect estimates, obtained using mixed models, are usually more reliable than the direct survey estimates. To draw inferences from mixed models, one can use Bayesian or frequentist approach. We consider the Bayesian approach in this dissertation. The Bayesian approach is straightforward. The prior and likelihood produce the posterior, which is used for all inferential purposes. It overcomes some of the shortcomings of the empirical Bayes approach. For example, the posterior variance automatically captures all sources of uncertainties in estimating small area parameters. But this approach requires the specification of a subjective prior on the model parameters. Moreover, in almost all situation, the posterior moments involve multi-dimensional integration and consequently closed form expressions cannot be obtained. To overcome the computational difficulties one needs to apply computer intensive MCMC methods. We apply linear mixed normal models (area level and unit level) to draw inferences for small areas when the variable of interest is continuous. We propose and evaluate a new prior distribution for the variance component. We use Laplace approximation to obtain accurate approximations to the posterior moments. The approximations present the Bayesian methodology in a transparent way, which facilitates the interpretation of the methodology to the data users. Our simulation study shows that the proposed prior yields good frequentist properties for the Bayes estimators relative to some other popular choices. This frequentist validation brings in an objective flavor to the so-called subjective Bayesian approach. The linear mixed models are, usually, not suitable for handling binary or count data, which are often encountered in surveys. To estimate the small area proportions, we propose a binomial-beta hierarchical model. Our formulation allows a regression specification and hence extends the usual exchangeable assumption at the second level. We carefully choose a prior for the shape parameter of the beta density. This new prior helps to avoid the extreme skewness present in the posterior distribution of the model parameters so that the Laplace approximation performs well.