Bayesian Estimators for Small Area Models when Auxiliary Information is Measured with Error

type="main" xml:id="sjos12120-abs-0001"> Small area estimators in linear models are typically expressed as a convex combination of direct estimators and synthetic estimators from a suitable model. When auxiliary information used in the model is measured with error, a new estimator, accounting for the measurement error in the covariates, has been proposed in the literature. Recently, for area-level model, Ybarra & Lohr (Biometrika, 95, 2008, 919) suggested a suitable modification to the estimates of small area means based on Fay & Herriot (J. Am. Stat. Assoc., 74, 1979, 269) model where some of the covariates are measured with error. They used a frequentist approach based on the method of moments. Adopting a Bayesian approach, we propose to rewrite the measurement error model as a hierarchical model; we use improper non-informative priors on the model parameters and show, under a mild condition, that the joint posterior distribution is proper and the marginal posterior distributions of the model parameters have finite variances. We conduct a simulation study exploring different scenarios. The Bayesian predictors we propose show smaller empirical mean squared errors than the frequentist predictors of Ybarra & Lohr (Biometrika, 95, 2008, 919), and they seem also to be more stable in terms of variability and bias. We apply the proposed methodology to two real examples.