Abstract
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 map** in Spanish provinces also indicate efficiency gains with respect to the conventional log-normal-nested error model used for poverty map**.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11749-018-0594-2/MediaObjects/11749_2018_594_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11749-018-0594-2/MediaObjects/11749_2018_594_Fig2_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11749-018-0594-2/MediaObjects/11749_2018_594_Fig3_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11749-018-0594-2/MediaObjects/11749_2018_594_Fig4_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11749-018-0594-2/MediaObjects/11749_2018_594_Fig5_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11749-018-0594-2/MediaObjects/11749_2018_594_Fig6_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11749-018-0594-2/MediaObjects/11749_2018_594_Fig7_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11749-018-0594-2/MediaObjects/11749_2018_594_Fig8_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11749-018-0594-2/MediaObjects/11749_2018_594_Fig9_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11749-018-0594-2/MediaObjects/11749_2018_594_Fig10_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11749-018-0594-2/MediaObjects/11749_2018_594_Fig11_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11749-018-0594-2/MediaObjects/11749_2018_594_Fig12_HTML.png)
Similar content being viewed by others
References
Battese GE, Harter RM, Fuller WA (1988) An error-components model for prediction of county crop areas using survey and satellite data. J Am Stat Assoc 83:28–36
Bordley RF, McDonald JB, Mantrala A (1996) Something new, something old: parametric models for the size distribution of income. J Income Distrib 6:91–103
Dastrup SR, Hartshorn R, McDonald JB (2007) The impact of taxes and transfer payments on the distribution of income: a parametric comparison. J Econ Inequal 5:353–369
Elbers C, Lanjouw JO, Lanjouw P (2003) Micro-level estimation of poverty and inequality. Econometrica 71(1):355–364
Ferretti C, Molina I (2011) Fast EB method for estimating complex poverty indicators in large populations. J Indian Soc Agric Stat 66:105–120
González-Manteiga W, Lombardía MJ, Molina I, Morales D, Santamaría L (2008) Bootstrap mean squared error of a small-area EBLUP. J Stat Comput Simul 78:443–462
González-Manteiga W, Lombardía MJ, Molina I, Morales D, Santamaría L (2008) Analytic and bootstrap approximations of prediction errors under a multivariate Fay–Herriot model. Comput Stat Data Anal 52:5242–5252
Graf M, Nedyalkova D (2013) Modeling of income and indicators of poverty and social exclusion using the generalized beta distribution of the second kind. Rev Income Wealth 60:821–842
Graf M, Nedyalkova D (2015) GB2: generalized beta distribution of the second kind: properties, likelihood, estimation. R package version 2.1.http://CRAN.R-project.org/package=GB2. Accessed 11 May 2015
Hansen CB, McDonald JB, Theodossiou P (2007) Some flexible parametric models for partially adaptive estimators of econometric models. Economics: the open-access. Open Assess E J 1:2007–7. https://doi.org/10.5018/economics-ejournal.ja.2007-7
Henningsen A, Toomet O (2011) maxLik: a package for maximum likelihood estimation in R. Comput Stat 26(3):443–458. https://doi.org/10.1007/s00180-010-0217-1 https://r-forge.r-project.org/projects/maxlik/
Hobza T, Morales D, Santamaría L (2018) Small area estimation of poverty proportions under unit-level temporal binomial–logit mixed models. TEST 27(2):270–294
Jenkins SP (2009) Distributionally-sensitive inequality indices and the GB2 income distribution. Rev Income Wealth 55:392–398
Kleiber C, Kotz S (2003) Statistical size distributions in economics and actuarial sciences. Wiley, Hoboken
Koenker R (2013) quantreg: quantile regression. R package version 5.05. http://CRAN.R-project.org/package=quantreg. Accessed 29 May 2018
Marhuenda Y, Molina I, Morales D, Rao JNK (2018) Poverty map** in small areas under a twofold nested error regression model. J R Stat Soc Ser A 180:1111–1136
McDonald J (1984) Some generalized functions for the size distribution of income. Econometrica 52(3):647–663
McDonald JB, Butler RJ (1987) Some generalized mixture distributions with an application to unemployment duration. Rev Econ Stat 69:232–240
McDonald JB, Xu YJ (1995) A generalization of the beta distribution with applications. J Econ 66:133–152
Molina I, Marhuenda Y (2015) sae: an R package for small area estimation. R J 7:81–98
Molina I, Rao JNK (2010) Small area estimation of poverty indicators. Can J Stat 38:369–385
Parker SC (1997) The distribution of self-employment income in the United Kingdom, 1976–1991. Econ J 107:455–466
Pfeffermann D, Sverchkov M (2007) Small-area estimation under informative probability sampling of areas and within the selected areas. J Am Stat Assoc 102:1427–1439
Pinheiro J, Bates D, DebRoy S, Sarkar D, R Core Team (2018) nlme: linear and nonlinear mixed effects models. R package version 3.1-131.1, https://CRAN.R-project.org/package=nlme. Accessed 7 Apr 2018
Rao JNK, Molina I (2015) Small area estimation, 2nd edn. Wiley, Hoboken
Rivest L-P, Verret F, Baillargeon S (2016) Unit level small area estimation with copulas. Can J Stat 44:397–415
Sepanski JH, Kong J (2008) A family of generalized beta distributions for income. Adv Appl Stat 10:75–84
Venter G (1983) Transformed beta and gamma distributions and aggregate losses. Proc Casualty Actuar Soc 70:156–193
Verret F, Rao JNK, Hidiroglou MA (2015) Model-based small area estimation under informative sampling. Survey Methodol 41:333–347
Yang X, Frees EW, Zhang Z (2011) A generalized beta-copula with applications in modeling multivariate long-tailed data. Insur Math Econ 49:265–284
You Y, Rao JNK (2002) A pseudo-empirical best linear unbiased prediction approach to small area estimation using survey weights. Can J Stat 30:431–439
Author information
Authors and Affiliations
Corresponding author
Additional information
This work has been supported by the Grants MTM2015-72907-EXP and MTM2015-69638-R (MINECO/FEDER, UE).
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Graf, M., Marín, J.M. & Molina, I. A generalized mixed model for skewed distributions applied to small area estimation. TEST 28, 565–597 (2019). https://doi.org/10.1007/s11749-018-0594-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11749-018-0594-2