  Estimating characteristics of domains (referred to as small areas) within a
population from sample surveys of the population is an important problem in
survey statistics. In this paper, we consider model-based small area estimation
under the nested error regression model. We discuss the construction of mixed
model estimators (empirical best linear unbiased predictors, EBLUPs) of small
area means and the conditional linear predictors of small area means. Under the
asymptotic framework of increasing numbers of small areas and increasing
numbers of units in each area, we establish asymptotic linearity results and
central limit theorems for these estimators which allow us to establish
asymptotic equivalences between estimators, approximate their sampling
distributions, obtain simple expressions for and construct simple estimators of
their asymptotic mean squared errors, and justify asymptotic prediction
intervals. We present model-based simulations that show that in quite small,
finite samples, our mean squared error estimator performs as well or better
than the widely-used \cite{prasad1990estimation} estimator and is much simpler,
so is easier to interpret. We also carry out a design-based simulation using
real data on consumer expenditure on fresh milk products to explore the
design-based properties of the mixed model estimators. We explain and interpret
some surprising simulation results through analysis of the population and
further design-based simulations. The simulations highlight important
differences between the model- and design-based properties of mixed model
estimators in small area estimation.
