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Keywords:

  • asymptotic approximation;
  • benchmarking;
  • best linear unbiased predictor;
  • binomial-beta model;
  • constrained Bayes;
  • empirical Bayes;
  • estimating equation;
  • generalized linear mixed model;
  • mean squared error;
  • natural exponential family;
  • parametric bootstrap;
  • mortality rates;
  • Poisson-gamma model;
  • second-order approximation;
  • small area estimation

ABSTRACT

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.