A Fay–Herriot model when auxiliary variables are measured with error (2019). Springer, 1-30.

Jan Pablo Burgard (Trier University), María Dolores Esteban (University Miguel Hernández of Elche), Domingo Morales (University Miguel Hernández of Elche) and Agustín Pérez (University Miguel Hernández of Elche).
Abstract. The Fay–Herriot model is an area-level linear mixed model that is widely used for estimating the domain means of a given target variable. Under this model, the dependent variable is a direct estimator calculated by using the survey data and the auxiliary variables are true domain means obtained from external data sources. Administrative registers do not always give good auxiliary variables so that statisticians sometimes take them from alternative surveys and therefore they are measured with error. We introduce a variant of the Fay–Herriot model that takes into account the measurement error of the auxiliary variables and give two fitting algorithms that calculate maximum and residual maximum likelihood estimates of the model parameters. Based on the new model, empirical best predictors of domain means are introduced and an approximation of its mean squared error is derived. We finally give an application to estimate poverty proportions in the Spanish Living Condition Survey, with auxiliary information from the Spanish Labour Force Survey.
Keywords. Fay–Herriot model; Small area estimation; Measurement error; Monte Carlo simulation; Poverty proportion