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Biometrika 2005 92(1):183-196; doi:10.1093/biomet/92.1.183
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© 2005 Biometrika Trust

On measuring the variability of small area estimators under a basic area level model

Gauri Sankar Datta1, J. N. K. Rao2 and David Daniel Smith3

1 Department of Statistics, University of Georgia, Athens, Georgia 30602-1952, U.S.A. gauri{at}stat.uga.edu, 2 School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada jrao{at}math.carleton.ca, 3 Department of Mathematics, Tennessee Technological University, Cookeville, Tennessee 38505, U.S.A. ddsmith{at}tntech.edu

In this paper based on a basic area level model we obtain second-order accurate approximations to the mean squared error of model-based small area estimators, using the Fay & Herriot (1979) iterative method of estimating the model variance based on weighted residual sum of squares. We also obtain mean squared error estimators unbiased to second order. Based on simulations, we compare the finite-sample performance of our mean squared error estimators with those based on method-of-moments, maximum likelihood and residual maximum likelihood estimators of the model variance. Our results suggest that the Fay–Herriot method performs better, in terms of relative bias of mean squared error estimators, than the other methods across different combinations of number of areas, pattern of sampling variances and distribution of small area effects. We also derive a noninformative prior on the model parameters for which the posterior variance of a small area mean is second-order unbiased for the mean squared error. The posterior variance based on such a prior possesses both Bayesian and frequentist interpretations.

Key Words: Empirical best linear unbiased prediction; Hierarchical Bayes; Mean squared error

Received February 2003. Revised May 2004.


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