Abstract
The extension of the univariate Lorenz curve to higher dimensions is not an obvious task. In this chapter, using the definition proposed by Arnold (Pareto Distributions. International Co-operative Publishing House, Fairland (1983)), closed expressions for the bivariate Lorenz curve are given, assuming that the underlying bivariate income distribution belong to the class of bivariate distributions with given marginals described by Sarmanov (Doklady Sov. Math. 168, 596–599 (1966)) and Lee (Commun. Stat. A-Theory 25, 1207–1222 (1996)). The expression of the bivariate Lorenz curve can be easily interpreted as a convex linear combination of products of classical and concentrated Lorenz curves. A closed expression for the bivariate Gini index (Arnold, Majorization and the Lorenz durve. In: Lecture Notes in Statistics, vol. 43. Springer, New York (1987)) in terms of the classical and concentrated Gini indices of the marginal distributions is given. This index can be decomposed in two factors, corresponding to the equality within and between variables. A specific model Pareto marginal distributions is studied. Other aspects are briefly discussed.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Arnold, B.C.: Pareto Distributions. International Co-operative Publishing House, Fairland (1983)
Arnold, B.C.: Majorization and the Lorenz durve. In: Lecture Notes in Statistics, vol. 43. Springer, New York (1987)
Atkinson, A.B.: Multidimensional deprivation: contrasting social welfare and counting approaches. J. Econ. Inequal. 1, 51–65 (2003)
Atkinson, A.B., Bourguignon, F.: The comparison of multi-dimensioned distributions of economic status. Rev. Econ. Stud. 49, 183–201 (1982)
Bairamov, I., Kotz, S.: On a new family of positive quadrant dependent bivariate distributions. Int. Math. J. 3, 1247–1254 (2003)
Gastwirth, J.L.: A general definition of the Lorenz curve. Econometrica 39, 1037–1039 (1971)
Huang, J.S., Kotz, S.: Modifications of the Farlie–Gumbel–Morgenstern distributions a tough hill to climb. Metrika 49, 135–145 (1999)
Kakwani, N.C.: Applications of Lorenz curves in economic analysis. Econometrica 45, 719–728 (1977)
Kolm, S.C.: Multidimensional equalitarianisms. Q. J. Econ. 91, 1–13 (1977)
Koshevoy, G.: Multivariate Lorenz majorization. Soc. Choice Welf. 12, 93–102 (1995)
Koshevoy, G., Mosler, K.: The Lorenz zonoid of a multivariate distribution. J. Am. Stat. Assoc. 91, 873–882 (1996)
Koshevoy, G., Mosler, K.: Multivariate gini indices. J. Multivar. Anal. 60, 252–276 (1997)
Lee, M.L.T.: Properties of the Sarmanov family of bivariate distributions. Commun. Stat. A-Theory 25, 1207–1222 (1996)
Maasoumi, E.: The measurement and decomposition of multi-dimensional inequality. Econometrica 54, 991–997 (1986)
Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: Theory of Majorization and its Applications, 2nd edn. Springer, New York (2011)
Mosler, K.: Multivariate dispersion, central regions and depth: the lift zonoid approach. In: Lecture Notes Statistics, vol. 165. Springer, Berlin (2002)
Sarabia, J.M.: Parametric Lorenz curves: models and applications. In: Chotikapanich, D. (ed.) Modeling Income Distributions and Lorenz Curves. Series: Economic Studies in Inequality, Social Exclusion and Well-being, vol. 4, pp. 167–190. Springer, New York (2008)
Sarabia, J.M., Castillo, E., Pascual, M., Sarabia, M.: Bivariate income distributions with lognormal conditionals. J. Econ. Inequal. 5, 371–383 (2007)
Sarabia, J.M., Castillo, E., Slottje, D.: An ordered family of Lorenz curves. J. Econ. 91, 43–60 (1999)
Sarmanov, O.V.: Generalized normal correlation and two-dimensional frechet classes. Doklady Sov. Math. 168, 596–599 (1966)
Slottje, D.J.: Relative price changes and inequality in the size distribution of various components. J. Bus. Econ. Stat. 5, 19–26 (1987)
Taguchi, T.: On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two-dimensional case-I. Ann. Inst. Stat. Math. 24, 355–382 (1972)
Taguchi, T.: On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two-dimensional case-II. Ann. Inst. Stat. Math. 24, 599–619 (1972)
Taguchi, T.: On the structure of multivariate concentration—some relationships among the concentration surface and two variate mean difference and regressions. Comput. Stat. Data Anal. 6, 307–334 (1988)
Tsui, K.Y.: Multidimensional generalizations of the relative and absolute inequality indices: the Atkinson–Kolm–Sen approach. J. Econ. Theory 67, 251–265 (1995)
Tsui, K.Y.: Multidimensional inequality and multidimensional generalized entropy measures: an axiomatic derivation. Soc. Choice Welf. 16, 145–157 (1999)
Acknowledgements
The authors thank to Ministerio de Economía y Competitividad (project ECO2010-15455) and Ministerio de Educación (FPU AP-2010-4907) for partial support.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this paper
Cite this paper
Sarabia, J.M., Jordá, V. (2014). Bivariate Lorenz Curves Based on the Sarmanov–Lee Distribution. In: Melas, V., Mignani, S., Monari, P., Salmaso, L. (eds) Topics in Statistical Simulation. Springer Proceedings in Mathematics & Statistics, vol 114. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2104-1_44
Download citation
DOI: https://doi.org/10.1007/978-1-4939-2104-1_44
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-2103-4
Online ISBN: 978-1-4939-2104-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)