Estimation of Median Incomes of the American States: Bayesian Estimation of Means of Subpopulations
The Fay–Herriot model (J Am Stat Assoc 74:269–277, 1979) is an immensely popular model in small area estimation of means of some characteristics for many related subpopulations. This model regresses the subpopulation mean on a set of auxiliary variables to borrow strength from other subpopulations. The Fay–Herriot model uses a fixed regression coefficient vector but only a random intercept term to account for variability that cannot be explained by the non-random mean function. In some applications, the non-random regression coefficient often does not adequately account for the variability of the subpopulation means. To accommodate extra variability, we consider an extension of the random intercept model by treating some of the regression coefficients also as random. This model is referred to as the random regression coefficient model. For the flexible random regression coefficient model, we allow the intercept and some of the regression coefficients to randomly vary across states with a suitable normal distribution. We use a suitable noninformative prior for all the model parameters to conduct our Bayesian analysis. We establish propriety of the resulting posterior density function and generate Monte Carlo samples from this distribution to get point estimates and associated measures of accuracy of these estimates. We also construct relevant credible intervals of the small area means as another measure of uncertainty for the point estimates. The method is illustrated to predict four-person family median incomes for 1989 for the US states.
KeywordsArea-level model Current population survey Empirical Bayes Fay–Herriot model Hierarchical Bayes Noninformative Bayes Propriety of posterior density Random regression coefficient model Small area estimation Stein’s shrinkage estimation
This report is released to inform interested parties of ongoing research and to encourage discussion. The views expressed on statistical, methodological, technical, or operational issues are those of the author(s) and not necessarily those of the U.S. Census Bureau, or the University of Georgia.
- Datta, G. S., Fay, R. E., & Ghosh, M. (1991). Hierarchical and empirical multivariate Bayes analysis in small area estimation. In Proceedings of the Seventh Annual Research Conference of the Bureau of the Census (Vol. 7, pp. 63–79). US Department of Commerce, Bureau of the Census.Google Scholar
- Datta, G. S., Ghosh, M., Nangia, N., & Natarajan, K. (1996). Estimation of median income of four-person families: A Bayesian approach. In D. A. Berry, K. M. Chaloner, & J. K. Geweke (Eds.), Bayesian analysis in statistics and econometrics (Chap. 11) (pp. 129–140). New York: Wiley.Google Scholar
- Datta, G. S., & Lahiri, P. (2000). A unified measure of uncertainty of estimated best linear unbiased predictors in small area estimation problems. Statistica Sinica, 10, 613–627.Google Scholar
- Efron, B., & Morris, C. (1973). Stein’s estimation rule and its competitors—An empirical Bayes approach. Journal of the American Statistical Association, 68, 117–130.Google Scholar
- James, W., & Stein, C. (1961). Estimation with quadratic loss. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability (Vol. 1, pp. 361–379). University of California Press.Google Scholar
- Stan Development Team. (2018). RStan: The R interface to Stan. R Package Version, 2(17), 3.Google Scholar
- Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability (Vol. 1, pp. 197–206). University of California Press.Google Scholar