We derive the best predictive estimator (BPE) of the fixed parameters under two well-known small area models, the Fay-Herriot model and the nested-error regression model. This leads to a new prediction procedure, called observed best prediction (OBP), which is different from the empirical best linear unbiased prediction (EBLUP). We show that BPE is more reasonable than the traditional estimators derived from estimation considerations, such as maximum likelihood (ML) and restricted maximum likelihood (REML), if the main interest is estimation of small area means, which is a mixed-model prediction problem. We use both theoretical derivations and empirical studies to demonstrate that the OBP can significantly outperform EBLUP in terms of the mean squared prediction error (MSPE), if the underlying model is misspecified. On the other hand, when the underlying model is correctly specified, the overall predictive performance of the OBP is very similar to that of the EBLUP if the number of small areas is large. A general theory about OBP, including its exact MSPE comparison with the BLUP in the context of mixed-model prediction, and asymptotic behavior of the BPE, is developed. A real data example is considered. A supplementary appendix is available online.
Bibliographical noteFunding Information:
Jiming Jiang is Professor, Department of Statistics, University of California, Davis, Davis, CA 95616 (E-mail: firstname.lastname@example.org). Thuan Nguyen is Assistant Professor, Department of Public Health and Preventive Medicine, Oregon Health and Science University, Portland, OR 97239-3098. J. Sunil Rao is Professor, Department of Epidemiology and Public Health, University of Miami, Miami, FL 33136. Jiming Jiang is partially supported by the NSF grant DMS-0809127. J. Sunil Rao is partially supported by the NSF grant DMS-0806076. The research of all three authors are partially supported by the NIH grant R01-GM085205A1. The authors are grateful to an Associate Editor and two Referees for their constructive comments that have led to improvements of the article.
- Fay-herriot model
- Mean squared prediction error (MSPE)
- Model misspecification
- Nested-error regression model