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Small area estimation under a measurement error bivariate Fay–Herriot model

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  • Jan Pablo Burgard

    (Trier University)

  • María Dolores Esteban

    (University Miguel Hernández de Elche)

  • Domingo Morales

    (University Miguel Hernández de Elche)

  • Agustín Pérez

    (University Miguel Hernández de Elche)

Abstract

The bivariate Fay–Herriot model is an area-level linear mixed model that can be used for estimating the domain means of two correlated target variables. Under this model, the dependent variables are direct estimators calculated from 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 bivariate Fay–Herriot model that takes into account the measurement error of the auxiliary variables and we give fitting algorithms to estimate the model parameters. Based on the new model, we introduce empirical best predictors of domain means and we propose a parametric bootstrap procedure for estimating the mean squared error. We finally give an application to estimate poverty proportions and gaps in the Spanish Living Condition Survey, with auxiliary information from the Spanish Labour Force Survey.

Suggested Citation

  • Jan Pablo Burgard & María Dolores Esteban & Domingo Morales & Agustín Pérez, 2021. "Small area estimation under a measurement error bivariate Fay–Herriot model," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(1), pages 79-108, March.
    • Handle: RePEc:spr:stmapp:v:30:y:2021:i:1:d:10.1007_s10260-020-00515-9
    DOI: 10.1007/s10260-020-00515-9
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    References listed on IDEAS

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    1. Marhuenda, Yolanda & Molina, Isabel & Morales, Domingo, 2013. "Small area estimation with spatio-temporal Fay–Herriot models," Computational Statistics & Data Analysis, Elsevier, vol. 58(C), pages 308-325.
    2. Isabel Molina & Ayoub Saei & M. José Lombardía, 2007. "Small area estimates of labour force participation under a multinomial logit mixed model," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 170(4), pages 975-1000, October.
    3. Serena Arima & William R. Bell & Gauri S. Datta & Carolina Franco & Brunero Liseo, 2017. "Multivariate Fay–Herriot Bayesian estimation of small area means under functional measurement error," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 180(4), pages 1191-1209, October.
    4. Torabi, Mahmoud, 2013. "Likelihood inference in generalized linear mixed measurement error models," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 549-557.
    5. Silvia Polettini & Serena Arima, 2015. "Small area estimation with covariates perturbed for disclosure limitation," Statistica, Department of Statistics, University of Bologna, vol. 75(1), pages 57-72.
    6. Serena Arima & Gauri S. Datta & Brunero Liseo, 2015. "Bayesian Estimators for Small Area Models when Auxiliary Information is Measured with Error," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(2), pages 518-529, June.
    7. Malay Ghosh & Karabi Sinha & Dalho Kim, 2006. "Empirical and Hierarchical Bayesian Estimation in Finite Population Sampling under Structural Measurement Error Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(3), pages 591-608, September.
    8. Esther López-Vizcaíno & María José Lombardía & Domingo Morales, 2015. "Small area estimation of labour force indicators under a multinomial model with correlated time and area effects," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 178(3), pages 535-565, June.
    9. Lynn M. R. Ybarra & Sharon L. Lohr, 2008. "Small area estimation when auxiliary information is measured with error," Biometrika, Biometrika Trust, vol. 95(4), pages 919-931.
    10. González-Manteiga, W. & Lombardi­a, M.J. & Molina, I. & Morales, D. & Santamari­a, L., 2008. "Analytic and bootstrap approximations of prediction errors under a multivariate Fay-Herriot model," Computational Statistics & Data Analysis, Elsevier, vol. 52(12), pages 5242-5252, August.
    11. Mahmoud Torabi & Gauri S. Datta & J. N. K. Rao, 2009. "Empirical Bayes Estimation of Small Area Means under a Nested Error Linear Regression Model with Measurement Errors in the Covariates," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(2), pages 355-369, June.
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