Small Area Estimation With Uncertain Random Effects

Abstract

Random effects models play an important role in model-based small area estimation. Random effects account for any lack of fit of a regression model for the population means of small areas on a set of explanatory variables. In a recent article, Datta, Hall, and Mandal showed that if the random effects can be dispensed with via a suitable test, then the model parameters and the small area means may be estimated with substantially higher accuracy. The work of Datta, Hall, and Mandal is most useful when the number of small areas, m, is moderately large. For large m, the null hypothesis of no random effects will likely be rejected. Rejection of the null hypothesis is usually caused by a few large residuals signifying a departure of the direct estimator from the synthetic regression estimator. As a flexible alternative to the Fay–Herriot random effects model and the approach in Datta, Hall, and Mandal, in this article we consider a mixture model for random effects. It is reasonably expected that small areas with population means explained adequately by covariates have little model error, and the other areas with means not adequately explained by covariates will require a random component added to the regression model. This model is a useful alternative to the usual random effects model and the data determine the extent of lack of fit of the regression model for a particular small area, and include a random effect if needed. Unlike the Datta, Hall, and Mandal approach which recommends excluding random effects from all small areas if a test of null hypothesis of no random effects is not rejected, the present model is more flexible. We used this mixture model to estimate poverty ratios for 5–17-year-old-related children for the 50 U.S. states and Washington, DC. This application is motivated by the SAIPE project of the U.S. Census Bureau. We empirically evaluated the accuracy of the direct estimates and the estimates obtained from our mixture model and the Fay–Herriot random effects model. These empirical evaluations and a simulation study, in conjunction with a lower posterior variance of the new estimates, show that the new estimates are more accurate than both the frequentist and the Bayes estimates resulting from the standard Fay–Herriot model. Supplementary materials for this article are available online.

• Empirical best linear unbiased prediction
• Fay–Herriot model
• Finite mixture models
• Hierarchical Bayes
• Mixed effects
• SAIPE project

Additional information

Notes on contributors

Gauri Sankar Datta

Gauri Sankar Datta, Department of Statistics, University of Georgia, Athens, GA 30602 and Center for Statistical Research and Methodology, U.S. Census Bureau (Email: [email protected]). Abhyuday Mandal, Department of Statistics, University of Georgia, Athens, GA 30602 (E-mail: [email protected]). Disclaimer: This report is released to inform interested parties of research and to encourage discussion of work in progress. The views expressed are those of the authors and not necessarily those of the U.S. Census Bureau. The research of Gauri S. Datta was partially supported through the grants DMS-09-14603 from the National Science Foundation and H98230-11-1-0208 from the National Security Agency. The research of Abhyuday Mandal was partially supported through the grants DMS-09-05731 from the National Science Foundation and H98230-13-1-0251 from the National Security Agency. The authors are thankful to Dr. William R. Bell for providing and explaining the poverty data used in our application. They also thank Jerry Maples of Census Bureau for valuable comments.

Abhyuday Mandal

Gauri Sankar Datta, Department of Statistics, University of Georgia, Athens, GA 30602 and Center for Statistical Research and Methodology, U.S. Census Bureau (Email: [email protected]). Abhyuday Mandal, Department of Statistics, University of Georgia, Athens, GA 30602 (E-mail: [email protected]). Disclaimer: This report is released to inform interested parties of research and to encourage discussion of work in progress. The views expressed are those of the authors and not necessarily those of the U.S. Census Bureau. The research of Gauri S. Datta was partially supported through the grants DMS-09-14603 from the National Science Foundation and H98230-11-1-0208 from the National Security Agency. The research of Abhyuday Mandal was partially supported through the grants DMS-09-05731 from the National Science Foundation and H98230-13-1-0251 from the National Security Agency. The authors are thankful to Dr. William R. Bell for providing and explaining the poverty data used in our application. They also thank Jerry Maples of Census Bureau for valuable comments.