Small Area Estimation With Uncertain Random Effects

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.