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Parametric Lorenz Curves: Models and Applications

In: Modeling Income Distributions and Lorenz Curves

Author

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  • José María Sarabia

    (University of Cantabria)

Abstract

The Lorenz curve (LC) is an important instrument for analyzing the size of distribution of income or wealth and inequality. Finding an appropriate functional form is an important practical and theoretical problem. In this chapter we study parametric models for the LC and some important applications. The basic properties that a function should satisfy in order to be a genuine LC are discussed. Next, we study the different ways for generating parametric families of LCs, as well as some of their basic properties, including their relationship with the underlying income distribution function. The basic parametric models proposed in the literature are studied, including the Pareto, lognormal and other important families of LCs. Some general strategies to obtain extensions and generalizations of the basic parametric models are presented. One of the main applications of LCs is the study of inequality. We begin studying different measures of inequality together with their expressions in terms of the LC. These measures include the Gini index and some of their generalizations proposed by Kakwani (1980) and Yitzhaki (1983). Their corresponding expressions for the proposed parametric families of LCs will be obtained. The Lorenz ordering is also studied. The Lorenz ordering is a partial order that allows the comparison of two distributions when its corresponding LCs do not intersect. Some basic properties of this order are studied, including the effect of transformations, its relations with other partial orderings and their application to important parametric income distributions. The recent proposal of multivariate versions of the LC are studied. Finally, some applications of the Lorenz curve are presented.

Suggested Citation

  • José María Sarabia, 2008. "Parametric Lorenz Curves: Models and Applications," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 9, pages 167-190, Springer.
    • Handle: RePEc:spr:esichp:978-0-387-72796-7_9
    DOI: 10.1007/978-0-387-72796-7_9
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    Citations

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    Cited by:

    1. Edwin Fourrier-Nicolaï & Michel Lubrano, 2021. "Bayesian Inference for Parametric Growth Incidence Curves," Research on Economic Inequality, in: Research on Economic Inequality: Poverty, Inequality and Shocks, volume 29, pages 31-55, Emerald Group Publishing Limited.
    2. Gholamreza Hajargasht & William E. Griffiths, 2016. "Inference for Lorenz Curves," Department of Economics - Working Papers Series 2022, The University of Melbourne.
    3. José María Sarabia & Vanesa Jorda, 2020. "Lorenz Surfaces Based on the Sarmanov–Lee Distribution with Applications to Multidimensional Inequality in Well-Being," Mathematics, MDPI, vol. 8(11), pages 1-17, November.
    4. Sarabia, José María, 2008. "A general definition of the Leimkuhler curve," Journal of Informetrics, Elsevier, vol. 2(2), pages 156-163.
    5. Liang Frank Shao & Melanie Krause, 2020. "Rising mean incomes for whom?," PLOS ONE, Public Library of Science, vol. 15(12), pages 1-20, December.
    6. Sarabia, José María & Jordá, Vanesa, 2014. "Explicit expressions of the Pietra index for the generalized function for the size distribution of income," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 416(C), pages 582-595.
    7. Genya Kobayashi & Kazuhiko Kakamu, 2019. "Approximate Bayesian computation for Lorenz curves from grouped data," Computational Statistics, Springer, vol. 34(1), pages 253-279, March.
    8. Melanie Krause, 2014. "Parametric Lorenz Curves and the Modality of the Income Density Function," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 60(4), pages 905-929, December.
    9. Cirillo, Pasquale, 2013. "Are your data really Pareto distributed?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(23), pages 5947-5962.
    10. Amparo Ba'illo & Javier C'arcamo & Carlos Mora-Corral, 2021. "Extremal points of Lorenz curves and applications to inequality analysis," Papers 2103.03286, arXiv.org.
    11. Sarabia, José María & Prieto, Faustino & Trueba, Carmen & Jordá, Vanesa, 2013. "About the modified Gaussian family of income distributions with applications to individual incomes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(6), pages 1398-1408.
    12. Fontanari Andrea & Cirillo Pasquale & Oosterlee Cornelis W., 2020. "Lorenz-generated bivariate Archimedean copulas," Dependence Modeling, De Gruyter, vol. 8(1), pages 186-209, January.
    13. Abdulhakim A. Al-Babtain & Ibrahim Elbatal & Christophe Chesneau & Farrukh Jamal, 2020. "Box-Cox Gamma-G Family of Distributions: Theory and Applications," Mathematics, MDPI, vol. 8(10), pages 1-24, October.
    14. Seungjin Han, 2018. "Repercussions of Endogenous Fast Rising Top Inequality," Department of Economics Working Papers 2018-03, McMaster University, revised May 2018.
    15. Frank A. Cowell & Philippe Kerm, 2015. "Wealth Inequality: A Survey," Journal of Economic Surveys, Wiley Blackwell, vol. 29(4), pages 671-710, September.
    16. Preda, Vasile & Dedu, Silvia & Gheorghe, Carmen, 2015. "New classes of Lorenz curves by maximizing Tsallis entropy under mean and Gini equality and inequality constraints," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 436(C), pages 925-932.
    17. C. García & J. García Pérez & J. Dorp, 2011. "Modeling heavy-tailed, skewed and peaked uncertainty phenomena with bounded support," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 20(4), pages 463-486, November.
    18. Fontanari Andrea & Cirillo Pasquale & Oosterlee Cornelis W., 2020. "Lorenz-generated bivariate Archimedean copulas," Dependence Modeling, De Gruyter, vol. 8(1), pages 186-209, January.
    19. Greselin, Francesca & Zitikis, Ricardas, 2015. "Measuring economic inequality and risk: a unifying approach based on personal gambles, societal preferences and references," MPRA Paper 65892, University Library of Munich, Germany.
    20. Sarabia, José María & Prieto, Faustino & Sarabia, María, 2010. "Revisiting a functional form for the Lorenz curve," Economics Letters, Elsevier, vol. 107(2), pages 249-252, May.
    21. Francesca Greselin & Ričardas Zitikis, 2018. "From the Classical Gini Index of Income Inequality to a New Zenga-Type Relative Measure of Risk: A Modeller’s Perspective," Econometrics, MDPI, vol. 6(1), pages 1-20, January.

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