Small Area Estimation with a Lognormal Mixed Model under Informative Sampling

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The demand for reliable business statistics at disaggregated levels, such as industry classes, increased considerably in recent years. Owing to small sample sizes for some of the domains, design-based methods may not provide estimates with adequate precision. Hence, modelbased small area estimation techniques that increase the effective sample size by borrowing strength are needed. Business data are frequently characterised by skewed distributions, with a few large enterprises that account for the majority of the total for the variable of interest, for example turnover. Moreover, the relationship between the variable of interest and the auxiliary variables is often non-linear on the original scale. In many cases, a lognormal mixed model provides a reasonable approximation of this relationship. In this article, we extend the empirical best prediction (EBP) approach to compensate for informative sampling, by incorporating design information among the covariates via an augmented modelling approach. This gives rise to the EBP under the augmented model. We propose to select the augmenting variable based on a joint assessment of a measure of predictive accuracy and a check of the normality assumptions. Finally, we compare our approach with alternatives in a model-based simulation study under different informative sampling mechanisms.

Alfons, A., M. Templ, and P. Filzmoser. 2010. “An Object-Oriented Framework for Statistical Simulation: The R Package simFrame.” Journal of Statistical Software 37: 1–36. Doi:

Asparouhov, T. 2006. “General Multi-Level Modeling with Sampling Weights.” Communications in Statistics Theory and Methods 35: 439–460. Doi:

Battese, G.E., R.M. Harter, and W.A. Fuller. 1988. “An Error Component Model for Prediction of County Crop Areas Using Survey and Satellite Data.” Journal of the American Statistical Association 83: 28–36. Doi:

Berg, E. and H. Chandra. 2014. “Small Area Prediction for a Unit-Level Lognormal Model.” Computational Statistics & Data Analysis 78: 159–175. Doi:

Bernardini Papalia, R., C. Bruch, T. Enderle, S. Falorsi, A. Fasulo, E. Hernandez-Vazquez, M. Ferrante, J. Kolb, R. Münnich, S. Pacei, R. Priam, P. Righi, T. Schmid, N. Shlomo, F. Volk, T. Zimmermann, and S. Zins. 2013. Best Practice Recommendations on Variance Estimation and Small Area Estimation in Business Surveys. Technical report, SSH-CT-2010-244767-BLUE-ETS. Available at: (accessed on 12 September 2017).

Burgard, J.P., R. Münnich, and T. Zimmermann. 2014. “The Impact of Sampling Designs on Small Area Estimates for Business Data.” Journal of Official Statistics 30: 749–771. Doi:

Chandra, H. and R. Chambers. 2011. “Small Area Estimation under Transformation to Linearity.” Survey Methodology 37: 39–51.

Ferrante, M.R., C. Trivisano, and E. Fabrizi. 2016. “Bayesian Small Area Estimation Methods for Business Survey Statistics.” In Proceedings of the 60th World Statistics Congress of the International Statistical Institute, 26–31 July 2015, 86–91, Rio de Janeiro.

Hidiroglou, M.A. and P. Lavallee. 2009. “Sampling and Estimation in Business Surveys.” In Handbook of Statistics, Volume 29 A, edited by D. Pfeffermann and C.R. Rao, Chapter 17, 441–470. Elsevier.

Hidiroglou, M.A. and P. Smith. 2005. “Developing Small Area Estimates for Business Surveys at the ONS.” Statistics in Transition 7: 527–539.

Jiang, J., P. Lahiri, and S.-M. Wan. 2002. “A Unified Jackknife Theory for Empirical Best Prediction with M-Estimation.” The Annals of Statistics 30: 1782–1810. Doi:

Krieg, S., V. Blaess, and M. Smeets. 2012. “Small Area Estimation of Turnover of the Structural Business Survey.” Discussion paper 201203, Statistics Netherlands. Available at: (accessed on 12 September 2017).

Pfeffermann, D. 2013. “New Important Developments in Small Area Estimation.” Statistical Science 28: 40–68. Doi:

Pfeffermann, D. and C.R. Rao. 2009a. Handbook of Statistics: Sample Surveys: Design, Methods and Applications, Volume 29A. Elsevier.

Pfeffermann, D. and C.R. Rao. 2009b. Handbook of Statistics: Sample Surveys: Inference and Analysis, Volume 29A. Elsevier.

Pfeffermann, D. and M. Sverchkov. 2007. “Small-Area Estimation under Informative Probability Sampling of Areas and within the Selected Areas.” Journal of the American Statistical Association 102: 1427–1439. Doi:

Pfeffermann, D. and M. Sverchkov. 2009. “Inference under Informative Sampling.” In Handbook of Statistics, Volume 29B, edited by D. Pfeffermann and C.R. Rao, Chapter 39, 455–487. Elsevier.

Prasad, N.G.N. and J.N.K. Rao. 1999. “On Robust Small Area Estimation Using a Simple Random Effects Model.” Survey Methodology 25: 67–72.

Rao, J.N.K. and I. Molina. 2015. Small Area Estimation. Hoboken, NJ: John Wiley &Sons. Doi:10.1002/9781118735855.

Shapiro, S.S. and M.B. Wilk. 1965. “An Analysis of Variance Test for Normality (Complete Samples).” Biometrika 52: 591–611. Doi:

Skinner, C. 1994. “Sample Models and Weights.” In Proceedings of the Section on Survey Research Methods: American Statistical Association, 13–18 August 1994. 133–142. Toronto. Available at: (accessed on 12 September 2017).

Tillé, Y. 2006. Sampling Algorithms, Springer Series in Statistics. New York: Springer.

Vaida, F. and S. Blanchard. 2005. “Conditional Akaike Information for Mixed-Effects Models.” Biometrika 92: 351–370. Doi:

Valliant, R., A.H. Dorfman, and R.M. Royall. 2000. Finite Population Sampling and Inference: a Prediction Approach. John Wiley.

Verret, F., M.A. Hidiroglou, and J.N.K. Rao. 2015. “Model-Based Small Area Estimation under Informative Sampling.” Survey Methodology 41: 333–347.

You, Y. and J.N.K. Rao. 2002. “A Pseudo-Empirical Best Linear Unbiased Prediction Approach to Small Area Estimation Using Survey Weights.” The Canadian Journal of Statistics 30: 431–439. Doi:

Zimmermann, T. 2018. The Interplay between Sampling Design and Statistical Modelling in Small Area Estimation. PhD thesis, University of Trier.

Journal of Official Statistics

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