Piotr Sulewski https://orcid.org/0000-0002-0788-6567 , Marcin Szymkowiak https://orcid.org/0000-0003-3432-4364

© Piotr Sulewski, Marcin Szymkowiak. Artykuł udostępniony na licencji CC BY-SA 4.0


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In income modelling studies, such well-known distributions as the Dagum, the lognormal or the Zenga distributions are often used as approximations of the observed distributions. The objective of the research described in the article is to verify the possibility of using other type of distributions, i.e. asymmetric distributions derived from normal distribution (ND) in the context of income modelling. Data from the 2011 EU-SILC survey on the monthly gross income per capita in Poland were used to assess the most important characteristics of the discussed distributions. The probability distributions were divided into two groups: I – distributions commonly used for income modelling (e.g. the Dagum distribution) and II – distributions derived from ND (e.g. the SU Johnson distribution). In addition to the visual evaluation of the usefulness of the analysed probability distributions, various numerical criteria were applied: information criteria for econometric models (such as the Akaike Information Criterion, Schwarz’s Bayesian Information Criterion and the Hannan-Quinn Information Criterion), measures of agreement, as well as empirical and theoretical characteristics, including a measure based on quantiles, specifically defined by the authors for the purposes of this article. The research found that the SU Johnson distribution (Group II), similarly to the Dagum distribution (Group I), can be successfully used for income modelling.


income modelling, EU-SILC, normal distribution, SU Johnson distribution, Dagum distribution


C13, C15, C55, D31


Aitchison, J., & Brown, J. A. C. (1957). The Lognormal Distribution: with special reference to its uses in economics. Cambridge University Press.

Arellano-Valle, R. B., Gómez, H. W., & Quintana, F. A. (2004). A new class of skew-normal distributions. Communications in Statistics – Theory and Methods, 33(7), 1465–1480. https://doi.org/10.1081/STA-120037254.

Azzalini, A. (1985). A Class of Distributions which Includes the Normal Ones. Scandinavian Journal of Statistics, 12(2), 171–178.

Bahrami, W., & Qasemi, E. (2015). A Flexible Skew-Generalized Normal Distribution. Journal of Statistical Research of Iran JSRI, 11(2), 131–145. https://doi.org/10.18869/acadpub.jsri.11.2.131.

Behboodian, J. (1970). On the modes of a mixture of two normal distributions. Technometrics, 12(1), 131–139. https://doi.org/10.2307/1267357.

Birnbaum, Z. W., & Saunders, S. C. (1969). A new family of life distributions. Journal of Applied Probability, 6(2), 637–652. https://doi.org/10.2307/3212003.

Brzeziński, M. (2013). Parametric Modelling of Income Distribution in Central and Eastern Europe. Central European Journal of Economic Modelling and Econometrics, 5(3), 207–230. https://doi.org/10.24425/cejeme.2013.119261.

Choudhury, K., & Abdul, M. M. (2011). Extended skew generalized normal distribution. METRON, 69(3), 265–278. https://doi.org/10.1007/BF03263561.

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81(7), 883–898. https://doi.org/10.1080/00949650903530745.

Dagum, C. (1977). A New Model of Personal Income Distribution: Specification and Estimation. Économie Appliquée, 30(3), 413–436.

Gaddum, J. H. (1945). Lognormal distributions. Nature, 156, 463–466. https://doi.org/10.1038/156463a0.

Główny Urząd Statystyczny. (2021). Dochody i warunki życia ludności Polski – raport z badania EU-SILC 2019. https://stat.gov.pl/en/topics/living-conditions/living-conditions/incomes-and-living-conditions-of-the-population-in-poland-report-from-the-eu-silc-survey-of-2019,1,12.html.

Gómez, H. W., Elal-Olivero, D., Salinas, H. S., & Bolfarine, H. (2011). Bimodal Extension Based on the Skew-Normal Distribution with Application to Pollen Data. Environmetrics, 22(1), 50–62. https://doi.org/10.1002/env.1026.

Gupta, R. C., & Gupta, R. D. (2008). Analyzing skewed data by power normal model. TEST, 17(1), 197–210. https://doi.org/10.1007/s11749-006-0030-x.

Jędrzejczak, A. (1993). Application of the Dagum distribution in the analysis of income distributions in Poland. Acta Universitatis Lodziensis. Folia Oeconomica, (131), 103–112.

Jędrzejczak, A. (2006). The characteristic of theoretical income distributions and their application to the analysis of wage distributions in Poland by regions. Acta Universitatis Lodziensis. Folia Oeconomica, (196), 183–198.

Jędrzejczak, A., & Pekasiewicz, D. (2020). Teoretyczne rozkłady dochodów gospodarstw domowych i ich estymacja. Wydawnictwo Uniwersytetu Łódzkiego.

Jędrzejczak, A., & Trzcińska, K. (2018). Application of the Zenga distribution to the analysis of household income in Poland by socio-economic group. Statistica & Applicazioni, 16(2), 123– 140. https://doi.org/10.26350/999999_000015.

Johnson, N. L. (1949). System of frequency curves generated by methods of translation. Biometrika, 36(1/2), 149–176. https://doi.org/10.2307/2332539.

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions (vol. 2, 2nd ed.). John Wiley & Sons.

Kim, H.-J. (2005). On a class of two-piece skew-normal distributions. Statistics. A Journal of Theoretical and Applied Statistics, 39(6), 537–553. https://doi.org/10.1080/02331880500366027.

Kordos, J. (1968). Metody matematyczne badania i analizy rozkładów dochodów ludności. Główny Urząd Statystyczny.

Kordos J. (1973). Metody analizy i prognozowania rozkładów płac i dochodów ludności. Państwowe Wydawnictwo Ekonomiczne.

Kot, S. M. (Ed.). (1999). Analiza ekonometryczna kształtowania się płac w Polsce w okresie transformacji. Wydawnictwo Naukowe PWN.

Kot, S. M. (2000). Ekonometryczne modele dobrobytu. Wydawnictwo Naukowe PWN.

Kumar, C. S., & Anusree, M. R. (2015). On an extended version of skew generalized normal distribution and some of its properties. Communications in Statistics – Theory and Methods, 44(3), 573–586. https://doi.org/10.1080/03610926.2012.739251.

Kunte, S., & Gore, A. P. (1992). The paradox of large samples. Current Science, 62(5), 393–395.

Lange, O. (1967). Wstęp do ekonometrii (4th ed.). Państwowe Wydawnictwo Naukowe.

Łukasiewicz, P., & Orłowski, A. (2004). Probabilistic Models of Income Distributions. Physica A: Statistical Mechanics and its Applications, 344(1–2), 146–151. https://doi.org/10.1016/j.physa.2004.06.106.

Ma, Y., & Genton, M. G. (2004). Flexible Class of Skew-Symmetric Distributions. Scandinavian Journal of Statistics, 31(3), 459–468.

McDonald, J. B. (1984). Some generalized functions for the size distribution of income. Econometrica, 52(3), 647–663. https://doi.org/10.2307/1913469.

Mead, M., Nassar, M. M., & Dey, S. (2018). A Generalization of Generalized Gamma Distributions. Pakistan Journal of Statistics and Operation Research, 14(1), 121–138. https://doi.org/10.18187/pjsor.v14i1.1692.

Metcalf, C. E. (1972). An Econometric Model of Income Distribution. Markham Publishing Company.

Nekoukhou, V., Alamatsaz, M. H., & Aghajani, A. H. (2013). A flexible skew-generalized normal distribution. Communications in Statistics – Theory and Methods, 42(13), 2324–2334. https://doi.org/10.1080/03610926.2011.599003.

Ostasiewicz, K. (2013). Adekwatność wybranych rozkładów teoretycznych dochodów w zależności od metody aproksymacji. Przegląd Statystyczny. Statistical Review, 60(4), 499–522.

Pareto, V. (1895). La legge della domanda. Giornale Degli Economisti, 10(6), 59–68.

Pratesi, M. (Ed.). (2016). Analysis of Poverty Data by Small Area Estimation. John Wiley & Sons. https://doi.org/10.1002/9781118814963.

R Core Team. (2021). The R Project for Statistical Computing. https://www.R-project.org/.

Rasekhi, M., Hamedani, G. G., & Chinipardaz, R. (2017). A flexible extension of skew generalized normal distribution. METRON, 75(1), 87–107. https://doi.org/10.1007/s40300-017-0106-2.

Rieck, J. R., & Nedelman, J. R. (1991). A Log-Linear Model for the Birnbaum-Saunders Distribution. Technometrics, 33(1), 51–60. https://doi.org/10.2307/1269007.

Salamaga, M. (2016). Badanie wpływu metody estymacji teoretycznych modeli rozkładu dochodów na jakość aproksymacji rozkładu dochodów mieszkańców Krakowa. Zeszyty Naukowe UEK, 3(951), 63–79. https://doi.org/10.15678/ZNUEK.2016.0951.0305.

Sharafi, M., & Behboodian, J. (2008). The Balakrishnan skew-normal density. Statistical Papers, 49(4), 769–778. https://doi.org/10.1007/s00362-006-0038-z.

Singh, S. K., & Maddala, G. S. (1976). A Function for Size Distribution of Income. Econometrica, 44(5), 963–970. https://doi.org/10.2307/1911538.

Stacy, E. W. (1962). A generalization of the gamma distribution. The Annals of Mathematical Statistics, 33(3), 1187–1192.

Stacy, E. W., & Mihram, G. A. (1965). Parameter Estimation for a Generalized Gamma Distribution. Technometrics, 7(3), 349–358. https://doi.org/10.2307/1266594.

Sulewski, P. (2019). Modified Lilliefors Goodness-of-fit Test for Normality. Communications in Statistics – Simulation and Computation, 51(3), 1199–1219. https://doi.org/10.1080/03610918.2019.1664580.

Sulewski, P. (in press). New Members of The Johnson Family of Probability Distributions: Properties and Application. REVSTAT – Statistical Journal.

Trzcińska, K. (2020). Analysis of Household Income in Poland Based on the Zenga Distribution and Selected Income Inequality Measure. Folia Oeconomica Stetinensia, 20(1), 421–436. https://doi.org/10.2478/foli-2020-0025.

Trzcińska, K. (2022). An Analysis of Household Income in Poland and Slovakia Based on Selected Income Models. Folia Oeconomica Stetinensia, 22(1), 287–301. https://doi.org/10.2478/foli-2022-0014.

Venegas, O., Sanhueza, A. I., & Gómez, H. W. (2011). An extension of the skew-generalized normal distribution and its derivation. Proyecciones. Journal of Mathematics, 30(3), 401–413. https://doi.org/10.4067/S0716-09172011000300007.

Vielrose, E. (1960). Rozkład dochodów według wielkości. Polskie Wydawnictwo Gospodarcze.

Wałęga, A., & Wałęga, G. (2021). Self-employment and over-indebtedness in Poland: Modelling income and debt repayments distribution. EBER Entrepreneurial Business and Economics Review, 9(4), 51–65. https://doi.org/10.15678/EBER.2021.090404.

Wiśniewski, J. (1934). Rozkład dochodów według wysokości. Instytut Badania Koniunktur Gospodarczych i Cen.

Yadegari, I., Gerami, A., & Khaledi, M. J. (2008). A generalization of the Balakrishnan skewnormal distribution. Statistics & Probability Letters, 78(10), 1165–1167. https://doi.org/10.1016/j.spl.2007.12.001.

Zenga, M. M. (2010). Mixture of Polisicchio’s truncated Pareto distributions with beta weights. Statistica & Applicazioni, 8(1), 3–25. https://www.vitaepensiero.it/scheda-articolo_digital/michele-zenga/mixture-of-polisicchios-truncated-pareto-distributions-with-beta-weights-999999_2010_0001_0002-151302.html.

Zenga, M. M., Pasquazzi, L., & Zenga, M. (2012). First applications of a new three-parameter distribution for non-negative variables. Statistica & Applicazioni, 10(2), 131–149. https://statisticaeapplicazioni.vitaepensiero.it/scheda-articolo_digital/leo-pasquazzi-mariangela-zenga-michele-zenga/first-applications-of-a-new-three-parameter-distribution-for-non-negative-variables-999999_2012_0002_0037-151352.html.

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