, Volume 28, Issue 2, pp 565–597 | Cite as

A generalized mixed model for skewed distributions applied to small area estimation

  • Monique Graf
  • J. Miguel MarínEmail author
  • Isabel Molina
Original Paper


Models with random (or mixed) effects are commonly used for panel data, in microarrays, small area estimation and many other applications. When the variable of interest is continuous, normality is commonly assumed, either in the original scale or after some transformation. However, the normal distribution is not always well suited for modeling data on certain variables, such as those found in Econometrics, which often show skewness even at the log scale. Finding the correct transformation to achieve normality is not straightforward since the true distribution is not known in practice. As an alternative, we propose to consider a much more flexible distribution called generalized beta of the second kind (GB2). The GB2 distribution contains four parameters with two of them controlling the shape of each tail, which makes it very flexible to accommodate different forms of skewness. Based on a multivariate extension of the GB2 distribution, we propose a new model with random effects designed for skewed response variables that extends the usual log-normal-nested error model. Under this new model, we find empirical best predictors of linear and nonlinear characteristics, including poverty indicators, in small areas. Simulation studies illustrate the good properties, in terms of bias and efficiency, of the estimators based on the proposed multivariate GB2 model. Results from an application to poverty mapping in Spanish provinces also indicate efficiency gains with respect to the conventional log-normal-nested error model used for poverty mapping.


Bootstrap Empirical best Mixed models Monte Carlo simulation Random effects 

Mathematics Subject Classification

62D05 62E99 62G09 

Supplementary material

11749_2018_594_MOESM1_ESM.pdf (191 kb)
Supplementary material 1 (pdf 191 KB)


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Copyright information

© Sociedad de Estadística e Investigación Operativa 2018

Authors and Affiliations

  • Monique Graf
    • 1
    • 2
  • J. Miguel Marín
    • 3
    Email author
  • Isabel Molina
    • 3
  1. 1.Institut de Statistique, Université de NeuchâtelNeuchâtelSwitzerland
  2. 2.Elpacos StatisticsLa NeuvevilleSwitzerland
  3. 3.Department of StatisticsUniversidad Carlos III de MadridGetafeSpain

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