Multivariate Fay–Herriot Bayesian estimation of small area means under functional measurement error

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.

[1]  M. Ghosh,et al.  Estimation of Median Income of Four-Person Families: A Bayesian Time Series Approach , 1996 .

[2]  E. B. Wilson,et al.  The Distribution of Chi-Square. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[3]  T. Maiti,et al.  MULTIVARIATE BAYESIAN SMALL AREA ESTIMATION : AN APPLICATION TO SURVEY ANDSATELLITE DATA , 1998 .

[4]  Rachel M. Harter,et al.  An Error-Components Model for Prediction of County Crop Areas Using Survey and Satellite Data , 1988 .

[5]  W. Bell,et al.  Applying Bivariate Binomial/Logit Normal Models to Small Area Estimation , 2013 .

[6]  Bradley P. Carlin,et al.  On MCMC sampling in hierarchical longitudinal models , 1999, Stat. Comput..

[7]  Serena Arima,et al.  Bayesian Estimators for Small Area Models when Auxiliary Information is Measured with Error , 2015 .

[8]  I. Basawa,et al.  Empirical best linear unbiased and empirical Bayes prediction in multivariate small area estimation , 1999 .

[9]  Malay Ghosh,et al.  Empirical Bayes estimation in finite population sampling under functional measurement error models , 2007 .

[10]  J. Rao,et al.  Empirical Bayes Estimation of Small Area Means under a Nested Error Linear Regression Model with Measurement Errors in the Covariates , 2009 .

[11]  S. Lohr,et al.  Small area estimation when auxiliary information is measured with error , 2008 .

[12]  M. Ghosh,et al.  Empirical and Hierarchical Bayesian Estimation in Finite Population Sampling under Structural Measurement Error Models , 2006 .

[13]  An Empirical Study on Using Previous American Community Survey Data Versus Census 2000 Data in SAIPE Models for Poverty Estimates , 2012 .

[14]  Mahmoud Torabi,et al.  Pseudo-empirical Bayes estimation of small area means under a nested error linear regression model with functional measurement errors , 2010 .

[15]  William R. Bell,et al.  Use of ACS Data to Produce SAIPE Model-Based Estimates of Poverty for Counties , 2007 .

[16]  R. Fay,et al.  Estimates of Income for Small Places: An Application of James-Stein Procedures to Census Data , 1979 .

[17]  Serena Arima,et al.  Objective Bayesian analysis of a measurement error small area model , 2012 .

[18]  R. Wolfinger,et al.  Nonconjugate Bayesian Analysis of Variance Component Models , 2000, Biometrics.