Multivariate Fay–Herriot Bayesian Estimation of Small Area Means Under Functional Measurement Error

Abstract

Summary Area level models, such as the Fay–Herriot model, aim to improve direct survey estimates for small areas by borrowing strength from related covariates and from direct estimates across all areas. In their multivariate form, where related population characteristics are jointly modelled, area level models allow for inference about functions of two or more characteristics and may exploit dependence between the response variables to improve small area predictions. When model covariates are observed with random error, such as those drawn from another survey, it is important to account for this error in the modelling. We present a Bayesian analysis of a multivariate Fay–Herriot model with functional measurement error, allowing for both joint modelling of related characteristics and accounting for random observation error in some of the covariates. We apply it to modelling 2010 and 2011 poverty rates of school-aged children for US counties, for predicting 2011 poverty rates and the 2010–2011 changes. For this application, the measurement error model results in great improvements in prediction when compared with the direct estimates, and ignoring the measurement error results in uncertainty estimates that are misleading. We propose a computational approach to implementing this model via an independence chain Markov chain Monte Carlo algorithm and prove the propriety of the posterior distribution under a class of non-informative priors.