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On Conditional Mean Squared Errors of Empirical Bayes Estimators in Mixed Models with Application to Small Area Estimation

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  • Shonosuke Sugasawa

    (Graduate School of Economics, University of Tokyo)

  • Tatsuya Kubokawa

    (Faculty of Economics, The University of Tokyo)

Abstract

This paper is concerned with the prediction of the conditional mean which involves the fixed and random effects based on the natural exponential family with a quadratic variance function. The best predictor is interpreted as the Bayes estimator in the Bayesian context, and the empirical Bayes estimator (EB) is useful for small area estimation in the sense of increasing precision of prediction for small area means. When data of the small area of interest are observed and one wants to know the prediction error of the EB based on the data, the conditional mean squared error (cMSE) given the data is used instead of the conventional unconditional MSE. The difference between the two kinds of MSEs is small and appears in the second-order terms in the classical normal theory mixed model. However, it is shown that the difference appears in the first-order or leading terms for distributions far from normality. Especially, the leading term in the cMSE is a quadratic concave function of the direct estimate in the small area for the binomial-beta mixed model, and an increasing function for the the Poisson-gamma mixed model, while the leading terms in the unconditional MSEs are constants for the two mixed models. Second-order unbiased estimators of the cMSE are provided in two ways based on the analytical and parametric bootstrap methods. Finally, the performances of the EB and the estimator of cMSE are examined through simulation and empirical studies.

Suggested Citation

  • Shonosuke Sugasawa & Tatsuya Kubokawa, 2014. "On Conditional Mean Squared Errors of Empirical Bayes Estimators in Mixed Models with Application to Small Area Estimation," CIRJE F-Series CIRJE-F-934, CIRJE, Faculty of Economics, University of Tokyo.
    • Handle: RePEc:tky:fseres:2014cf934
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    References listed on IDEAS

    as
    1. Gauri Sankar Datta & J. N. K. Rao & David Daniel Smith, 2005. "On measuring the variability of small area estimators under a basic area level model," Biometrika, Biometrika Trust, vol. 92(1), pages 183-196, March.
    2. Tatsuya Kubokawa & Mana Hasukawa & Kunihiko Takahashi, 2014. "On Measuring Uncertainty of Benchmarked Predictors with Application to Disease Risk Estimate," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(2), pages 394-413, June.
    3. Malay Ghosh & Tapabrata Maiti, 2008. "Empirical Bayes Confidence Intervals for Means of Natural Exponential Family‐Quadratic Variance Function Distributions with Application to Small Area Estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 484-495, September.
    4. Gauri Datta & Tatsuya Kubokawa & Isabel Molina & J. Rao, 2011. "Estimation of mean squared error of model-based small area estimators," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(2), pages 367-388, August.
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