"Linear Mixed Models and Small Area Estimation"(in Japanese)

Sample survey data can be used to derive a reliable estimate of a total mean for a large area. When the same data are used to estimate means of small areas like city, county or town belonging to the large area, the usual direct estimators like the sample mean have unacceptably large standard errors due to the small sizes of the samples in the small areas. This is called a small area problem. To find more accurate estimates for given small areas, one needs to "borrow strength" from the related areas. The linear mixed model (LMM) is recognized as an appropriate model for handling such a problem, and the resulting empirical best linear unbiased predictor (EBLUP) can yield a smaller standard error. This article gives a review of the small area estimation based on LMM. Especially, the article explains how the structure of (common parameters)+(random effects) in LMM works to get accurate estimates. The estimators of the mean squared errors of EBLUP and the confidence interval based on EBLUP are derived to evaluate accuracy of EBLUP. Finally, some generalizations and various variants of LMM are described for analyzing spatial data, and the generalized linear mixed model (GLMM) and its application to estimation of mortality rates are explained.