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On Measuring Uncertainty of Benchmarked Predictors with Application to Disease Risk Estimate

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  • Tatsuya Kubokawa
  • Mana Hasukawa
  • Kunihiko Takahashi

Abstract

type="main" xml:id="sjos12040-abs-0001"> Empirical Bayes (EB) estimates in general linear mixed models are useful for the small area estimation in the sense of increasing precision of estimation of small area means. However, one potential difficulty of EB is that the overall estimate for a larger geographical area based on a (weighted) sum of EB estimates is not necessarily identical to the corresponding direct estimate such as the overall sample mean. Another difficulty is that EB estimates yield over-shrinking, which results in the sampling variance smaller than the posterior variance. One way to fix these problems is the benchmarking approach based on the constrained empirical Bayes (CEB) estimators, which satisfy the constraints that the aggregated mean and variance are identical to the requested values of mean and variance. In this paper, we treat the general mixed models, derive asymptotic approximations of the mean squared error (MSE) of CEB and provide second-order unbiased estimators of MSE based on the parametric bootstrap method. These results are applied to natural exponential families with quadratic variance functions. As a specific example, the Poisson-gamma model is dealt with, and it is illustrated that the CEB estimates and their MSE estimates work well through real mortality data.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:scjsta:v:41:y:2014:i:2:p:394-413
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    References listed on IDEAS

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    1. 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.
    2. G. Datta & M. Ghosh & R. Steorts & J. Maples, 2011. "Bayesian benchmarking with applications to small area estimation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(3), pages 574-588, November.
    3. Malay Ghosh, 2004. "Small-area estimation based on natural exponential family quadratic variance function models and survey weights," Biometrika, Biometrika Trust, vol. 91(1), pages 95-112, March.
    4. Frey, Jesse & Cressie, Noel, 2003. "Some results on constrained Bayes estimators," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 389-399, December.
    5. Tatsuya Kubokawa & Mana Hasukawa & Kunihiko Takahashi, 2012. "On Measuring Uncertainty of Benchmarked Predictors with Application to Disease Risk Estimatee," CIRJE F-Series CIRJE-F-861, CIRJE, Faculty of Economics, University of Tokyo.
    6. Tatsuya Kubokawa, 2012. "Mixed Effects Prediction under Benchmarking and Applications to Small Area Estimation," CIRJE F-Series CIRJE-F-832, CIRJE, Faculty of Economics, University of Tokyo.
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    Cited by:

    1. 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.
    2. Sugasawa, Shonosuke & Kubokawa, Tatsuya, 2016. "On conditional prediction errors in mixed models with application to small area estimation," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 18-33.

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