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On Measuring Uncertainty of Small Area Estimators with Higher Order Accuracy

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  • Tatsuya Kubokawa

    (Faculty of Economics, University of Tokyo)

Abstract

The empirical best linear unbiased predictor (EBLUP) or the empirical Bayes estimator (EB) in the linear mixed model is recognized useful for the small area estimation, because it can increase the estimation precision by using the information from the related areas. Two of the measures of uncertainty of EBLUP is the estimation of the mean squared error (MSE) and the confidence interval, which have been studied under the second-order accuracy in the literature. This paper provides the general analytical results for these two measures in the unified framework, namely, we derive the conditions on the general consistent estimators of the variance components to satisfy the third-order accuracy in the MSE estimation and the confidence interval in the general linear mixed normal models. Those conditions are shown to be satisfied by not only the maximum likelihood (ML) and restricted maximum likelihood (REML), but also the other estimators including the Prasad-Rao and Fay-Herriot estimators in specific models.

Suggested Citation

  • Tatsuya Kubokawa, 2010. "On Measuring Uncertainty of Small Area Estimators with Higher Order Accuracy," CIRJE F-Series CIRJE-F-754, CIRJE, Faculty of Economics, University of Tokyo.
    • Handle: RePEc:tky:fseres:2010cf754
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    References listed on IDEAS

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    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. Cressie, N. & Lahiri, S. N., 1993. "The Asymptotic Distribution of REML Estimators," Journal of Multivariate Analysis, Elsevier, vol. 45(2), pages 217-233, May.
    3. Basu, Ruma & Ghosh, J. K. & Mukerjee, Rahul, 2003. "Empirical Bayes prediction intervals in a normal regression model: higher order asymptotics," Statistics & Probability Letters, Elsevier, vol. 63(2), pages 197-203, June.
    4. Gauri Sankar Datta & Malay Ghosh & David Daniel Smith & Parthasarathi Lahiri, 2002. "On an Asymptotic Theory of Conditional and Unconditional Coverage Probabilities of Empirical Bayes Confidence Intervals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 29(1), pages 139-152, March.
    5. Peter Hall & Tapabrata Maiti, 2006. "On parametric bootstrap methods for small area prediction," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(2), pages 221-238, April.
    6. Eric V. Slud & Tapabrata Maiti, 2006. "Mean‐squared error estimation in transformed Fay–Herriot models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(2), pages 239-257, April.
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    Cited by:

    1. Kubokawa, Tatsuya & Nagashima, Bui, 2012. "Parametric bootstrap methods for bias correction in linear mixed models," Journal of Multivariate Analysis, Elsevier, vol. 106(C), pages 1-16.

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