Pseudo‐empirical Bayes estimation of small area means based on James–Stein estimation in linear regression models with functional measurement error

Abstract

AbstractSmall area estimation plays an important role in making reliable inference for subpopulations (areas) for which relatively small samples or no samples are available. In model‐based small area estimation studies, linear and generalized linear mixed models have been used extensively assuming that covariates are not subjected to measurement errors. Recently, there have been studies considering this problem under the functional measurement error for covariates using the maximum likelihood method and the method of moments. In this paper, we study the James–Stein estimator of the true covariate subject to the functional measurement error. To this end, we obtain a new pseudo‐empirical Bayes (PEB) predictor of small area means based on the James–Stein estimator. Then, we show that the new PEB predictor is asymptotically optimal. The weighted and unweighted jackknife estimators of the mean squared prediction error of the new PEB predictor are also derived. Simulation studies are conducted to evaluate the performance of the proposed approach. We observe that the PEB predictor based on the James–Stein estimator performs better than those based on the maximum likelihood method and the method of moments. Finally, we apply the proposed methodology to a real dataset. The Canadian Journal of Statistics 43: 265–287; 2015 © 2015 Statistical Society of Canada