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.

A popular area level model used for the estimation of small area means is the Fay-Herriot model. This model involves unobservable random effects for the areas apart from the (fixed) linear regression based on area level covariates. Empirical best linear unbiased predictors of small area means are obtained by estimating the area random effects, and they can be expressed as a weighted average of area-specific direct estimators and regression-synthetic estimators. In some cases the observed data do not support the inclusion of the area random effects in the model. Excluding these area effects leads to the regression-synthetic estimator, that is, a zero weight is attached to the direct estimator. A preliminary test estimator of a small area mean obtained after testing for the presence of area random effects is studied. On the other hand, empirical best linear unbiased predictors of small area means that always give non-zero weights to the direct estimators in all areas together with alternative estimators based on the preliminary test are also studied. The preliminary testing procedure is also used to define new mean squared error estimators of the point estimators of small area means. Results of a limited simulation study show that, for small number of areas, the preliminary testing procedure leads to mean squared error estimators with considerably smaller average absolute relative bias than the usual mean squared error estimators, especially when the variance of the area effects is small relative to the sampling variances.

The present article discusses a model-based approach towards adjustment of the 1988 Census Dress Rehearsal Data collected from test sites in Missouri. The primary objective is to develop procedures that can be used to model data from the 1990 Census Post Enumeration Survey in April, 1991 and smooth survey-based estimates of the adjustment factors. We have proposed in this paper hierarchical Bayes (HB) and empirical Bayes (EB) procedures which meet this objective. The resulting estimators seem to improve consistently on the estimators of the adjustment factors based on dual system estimation (DSE) as well as the smoothed regression estimators.

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.

A popular area level model used for the estimation of small area means is the Fay-Herriot model. This model involves unobservable random effects for the areas apart from the (fixed) linear regression based on area level covariates. Empirical best linear unbiased predictors of small area means are obtained by estimating the area random effects, and they can be expressed as a weighted average of area-specific direct estimators and regression-synthetic estimators. In some cases the observed data do not support the inclusion of the area random effects in the model. Excluding these area effects leads to the regression-synthetic estimator, that is, a zero weight is attached to the direct estimator. A preliminary test estimator of a small area mean obtained after testing for the presence of area random effects is studied. On the other hand, empirical best linear unbiased predictors of small area means that always give non-zero weights to the direct estimators in all areas together with alternative estimators based on the preliminary test are also studied. The preliminary testing procedure is also used to define new mean squared error estimators of the point estimators of small area means. Results of a limited simulation study show that, for small number of areas, the preliminary testing procedure leads to mean squared error estimators with considerably smaller average absolute relative bias than the usual mean squared error estimators, especially when the variance of the area effects is small relative to the sampling variances.

The present article discusses a model-based approach towards adjustment of the 1988 Census Dress Rehearsal Data collected from test sites in Missouri. The primary objective is to develop procedures that can be used to model data from the 1990 Census Post Enumeration Survey in April, 1991 and smooth survey-based estimates of the adjustment factors. We have proposed in this paper hierarchical Bayes (HB) and empirical Bayes (EB) procedures which meet this objective. The resulting estimators seem to improve consistently on the estimators of the adjustment factors based on dual system estimation (DSE) as well as the smoothed regression estimators.