We study and partially solve three distinct problems in small area estimation. The problems are loosely connected by a common theme of prediction and (empirical) Bayesian models.

In the first part of the thesis we consider prediction in a survey small area context with spatially correlated errors. We introduce a novel asymptotic framework in which the spatially correlated small areas form clusters, the number of such clusters and the number of small areas in each cluster growing with sample size. Under such an asymptotic framework we show consistency and asymptotic normality of the parameter estimators. For empirical predictors based on model estimates, we show through simulation and a real data example, improved prediction over estimates ignoring spatial error-correlations.

The second part of the thesis involves using a hierarchical Bayes approach to solve the problem of multiple comparison in small area estimation. In the context of multiple comparison, a new class of moment matching priors is introduced. This class includes the well-known superharmonic prior due to Stein. Through data analysis and simulation we illustrate the use of our class of priors.

In the third part of the thesis, for a special case of the nested error regression model, we derive a non-parametric second order unbiased estimator of the mean squared error of the empirical best linear unbiased predictor. For the balanced case, the Prasad-Rao estimator is shown to be second order unbiased when the small area effects are non-normal. Through simulation we show that the Prasad-Rao estimator is robust for departures from normality.