Analysis

In many applications, the population means of geographically adjacent small areas exhibit a spatial variation. If available auxiliary variables do not adequately account for the spatial pattern, the residual variation will be included in the random effects. As a result, the independent and identical distribution assumption on random effects of the Fay-Herriot model will fail. Furthermore, limited resources often prevent numerous sub-populations from being included in the sample, resulting in non-sampled small areas. The problem can be exacerbated for predicting means of non-sampled small areas using the above Fay-Herriot model as the predictions will be made based solely on the auxiliary variables. To address such inadequacy, we consider Bayesian spatial random-effect models that can accommodate multiple non-sampled areas. Under mild conditions, we establish the propriety of the posterior distributions for various spatial models for a useful class of improper prior densities on model parameters. The effectiveness of these spatial models is assessed based on simulated and real data. Specifically, we examine predictions of statewide four-person family median incomes based on the 1990 Current Population Survey and the 1980 Census for the United States of America.

Small area estimation using area-level models can sometimes benefit from covariates that are observed subject to random errors, such as covariates that are themselves estimates drawn from another survey. Given estimates of the variances of these measurement (sampling) errors for each small area, one can account for the uncertainty in such covariates using measurement error models (e.g., Ybarra and Lohr, 2008). Two types of area-level measurement error models have been examined in the small area estimation literature. The functional measurement error model assumes that the underlying true values of the covariates with measurement error are fixed but unknown quantities. The structural measurement error model assumes that these true values follow a model, leading to a multivariate model for the covariates observed with error and the original dependent variable. We compare and contrast these two models with the alternative of simply ignoring measurement error when it is present (naïve model), exploring the consequences for prediction mean squared errors of use of an incorrect model under different underlying assumptions about the true model. Comparisons done using analytic formulas for the mean squared errors assuming model parameters are known yield some surprising results. We also illustrate results with a model fitted to data from the U.S. Census Bureau’s Small Area Income and Poverty Estimates (SAIPE) Program.

In many applications, the population means of geographically adjacent small areas exhibit a spatial variation. If available auxiliary variables do not adequately account for the spatial pattern, the residual variation will be included in the random effects. As a result, the independent and identical distribution assumption on random effects of the Fay-Herriot model will fail. Furthermore, limited resources often prevent numerous sub-populations from being included in the sample, resulting in non-sampled small areas. The problem can be exacerbated for predicting means of non-sampled small areas using the above Fay-Herriot model as the predictions will be made based solely on the auxiliary variables. To address such inadequacy, we consider Bayesian spatial random-effect models that can accommodate multiple non-sampled areas. Under mild conditions, we establish the propriety of the posterior distributions for various spatial models for a useful class of improper prior densities on model parameters. The effectiveness of these spatial models is assessed based on simulated and real data. Specifically, we examine predictions of statewide four-person family median incomes based on the 1990 Current Population Survey and the 1980 Census for the United States of America.

Small area estimation using area-level models can sometimes benefit from covariates that are observed subject to random errors, such as covariates that are themselves estimates drawn from another survey. Given estimates of the variances of these measurement (sampling) errors for each small area, one can account for the uncertainty in such covariates using measurement error models (e.g., Ybarra and Lohr, 2008). Two types of area-level measurement error models have been examined in the small area estimation literature. The functional measurement error model assumes that the underlying true values of the covariates with measurement error are fixed but unknown quantities. The structural measurement error model assumes that these true values follow a model, leading to a multivariate model for the covariates observed with error and the original dependent variable. We compare and contrast these two models with the alternative of simply ignoring measurement error when it is present (naïve model), exploring the consequences for prediction mean squared errors of use of an incorrect model under different underlying assumptions about the true model. Comparisons done using analytic formulas for the mean squared errors assuming model parameters are known yield some surprising results. We also illustrate results with a model fitted to data from the U.S. Census Bureau’s Small Area Income and Poverty Estimates (SAIPE) Program.