Selection of Prior for the Variance Component and Approximations for Posterior Moments in the Fay-Herriot Model

In the Fay-Herriot model, a prior distribution for the variance component allows posterior moments to be approximated with the Laplace method, avoiding computer intensive Monte Carlo Markov chains. The extremely skewed posterior distribution of the variance component results from the asymmetry of the parameter space with variance parameters constrained to be positive. The prior avoids the extreme skewness of the posterior in contrast to the commonly used uniform prior. With this prior, the mean squared error and coverage in the approximate hierarchical Bayes method are satisfactory when used to estimate small area means. Computation time is shorter than with Monte Carlo Markov chains. The approximations give easy interpretations of Bayesian methods and highlight frequentist properties of the parameters.