Unifying small area estimators based on area-level and unit-level models through calibration

When estimating area means, direct estimators based on area-specific data are usually consistent under the sampling design without model assumptions. However, they are inefficient if the area sample size is small. In small area estimation, model assumptions linking the areas are used to "borrow strength" from other areas. The basic area-level model provides design-consistent estimators, but error variances are assumed to be known. In practice, they are estimated with the (scarce) area-specific data. These estimators are inefficient, and their error is not accounted for in the associated mean-squared error estimators. The basic unit-level model does not require knowledge of the error variances, but it does not include the survey weights. We describe a unified estimator of an area mean that may be obtained both from an area-level model or a unit-level model and is based on consistent estimators of the model error variances as the number of areas increases. This unified predictor is obtained assuming that the unit-level model holds but accounts for the survey weights. We propose bootstrap mean-squared error estimators that account for the uncertainty due to the estimation of the error variances. In simulations, our new small area estimators and bootstrap mean squared error estimators perform better than alternatives. We apply the results to education data from Colombia.