  National statistical agencies are regularly required to produce estimates
about various subpopulations, formed by demographic and/or geographic
classifications, based on a limited number of samples. Traditional direct
estimates computed using only sampled data from individual subpopulations are
usually unreliable due to small sample sizes. Subpopulations with small samples
are termed small areas or small domains. To improve on the less reliable direct
estimates, model-based estimates, which borrow information from suitable
auxiliary variables, have been extensively proposed in the literature. However,
standard model-based estimates rely on the normality assumptions of the error
terms. In this research we propose a hierarchical Bayesian (HB) method for the
unit-level nested error regression model based on a normal mixture for the
unit-level error distribution. To implement our proposal we use a uniform prior
for the regression parameters, random effects variance parameter, and the
mixing proportion, and we use a partially proper non-informative prior
distribution for the two unit-level error variance components in the mixture.
We apply our method to two examples to predict summary characteristics of farm
products at the small area level. One of the examples is prediction of twelve
county-level crop areas cultivated for corn in some Iowa counties. The other
example involves total cash associated in farm operations in twenty-seven
farming regions in Australia. We compare predictions of small area
characteristics based on the proposed method with those obtained by applying
the Datta and Ghosh (1991) and the Chakraborty et al. (2018) HB methods. Our
simulation study comparing these three Bayesian methods showed the superiority
of our proposed method, measured by prediction mean squared error, coverage
probabilities and lengths of credible intervals for the small area means.
