The University of Georgia
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 Datta, Hall and Mandal (2011, JASA), we showed that if the random effects can be dispensed with through a statistical test, then the model parameters and the small area means can be estimated substantially accurately. This work 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 is usually due to a few large residuals signifying a departure of the direct estimator from the synthetic regression estimator. As a flexible alternative to the Fay-Herriot model and the approach in Datta et al. (2011), we consider a mixture model for random effects. Small areas whose population means are explained adequately by covariates have little model error, and the areas whose means lack adequate explanation will need a random component in the regression model. This model is a flexible alternative to the usual random effects model and the data determine if a small area needs a random effect. Unlike Datta et al. (2011) who recommend excluding random effects from all areas if a test of no random effects is not rejected, the present model is more flexible. We used this mixture model to estimate poverty ratios for 5- to 17-year old related children for the 50 U.S. states and Washington, DC. We empirically evaluated the accuracy of the direct estimates, the estimates obtained from our mixture model and those from the Fay-Herriot model. Empirical evaluations and simulations seem to indicate that the new estimates enjoy an edge over the estimates from the standard Fay-Herriot model.