The Canadian Journal of Statistics Vol. xx, No. yy, 2015, Pages 1–23 La revue canadienne de statistique 1 Pseudo-empirical Bayes estimation of small area means based on James–Stein estimation in linear regression models with functional measurement error Elaheh TORKASHVAND 1 , Mohammad JAFARI JOZANI 1 * and Mahmoud TORABI 2 1 Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 2 Department of Community Health Sciences, University of Manitoba, Winnipeg, Manitoba, Canada R3E 0W3 Key words and phrases: Empirical Bayes; Jackknife method; James–Stein estimator; mean squared predic- tion error; small area estimation. MSC 2010: Primary 62F10; secondary 62F30; tertiary 62C20 Abstract: Small 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 xx: 1–23; 2015 © 2015 Statistical Society of Canada R´ esum´ e: L’estimation pour petits domaines permet une inf´ erence fiable pour des sous-populations (do- maines) dans lesquelles le nombre de donn´ ees observ´ ees est petit, voire nul. Les m´ ethodes pour petits do- maines bas´ ees sur un mod` ele font un usage extensif de mod` eles lin´ eaires et de mod` eles lin´ eaires g´ en´ eralis´ es en supposant que les covariables ne comportent pas d’erreur de mesure. Des ´ etudes publi´ ees r´ ecemment reprennent les mˆ emes probl` emes avec un mod` ele d’erreur de mesure fonctionnelle en l’ajustant par le max- imum de vraisemblance ou la m´ ethode des moments. Les auteurs ´ etudient l’estimateur de James–Stein des vraies covariables sujettes ` a une erreur de mesure fonctionnelle. Ils obtiennent un nouvel estimateur bay´ esien pseudo-empirique pour petits domaines bas´ e sur l’estimateur de James–Stein et montrent que ce nouvel estimateur est asymptotiquement optimal. Ils calculent les estimateurs jackknife pond´ er´ es et non pond´ er´ es de l’erreur quadratique moyenne de pr´ evision. Les auteurs pr´ esentent des simulations pour ´ evaluer la performance de leur approche. Ils observent que la performance de l’estimateur propos´ e est sup´ erieure ` a celle des m´ ethodes bas´ ees sur le maximum de vraisemblance et la m´ ethode des moments. Ils appliquent aussi leur m´ ethodologie ` a un jeu de donn´ ees r´ eelles. La revue canadienne de statistique xx: 1–23; 2015 © 2015 Soci´ et´ e statistique du Canada * Author to whom correspondence may be addressed. E-mail: M Jafari Jozani@umanitoba.ca © 2015 Statistical Society of Canada / Soci´ et´ e statistique du Canada CJS 11245