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Multivariate Gini indices

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  • Koshevoy, Gleb
  • Mosler, Karl

Abstract

The Gini index and the Gini mean difference of a univariate distribution are extended to measure the disparity of a general d-variate distribution. We propose and investigate two approaches, one based on the distance of the distribution from itself, the other on the volume of a convex set in (d + 1)- space, named the lift zonoid of the distribution. When d = 1, this volume equals the area between the usual Lorenz curve and the line of zero disparity, up to a scale factor. We get two definitions of the multivariate Gini index, which are different (when d > 1) but connected through the notion of the lift zonoid. Both notions inherit properties of the univariate Gini index, in particular, they are vector scale invariant, continuous, bounded by 0 and 1, and the bounds are sharp. They vanish if and only if the distribution is concentrated at one point. The indices have a ceteris paribus property and are consistent with multivariate extensions of the Lorenz order. Illustrations with data conclude the paper.

Suggested Citation

  • Koshevoy, Gleb & Mosler, Karl, 1995. "Multivariate Gini indices," Discussion Papers in Econometrics and Statistics 7/95, University of Cologne, Institute of Econometrics and Statistics.
  • Handle: RePEc:zbw:ucdpse:9507
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    Cited by:

    1. Thibault Gajdos & John Weymark, 2005. "Multidimensional generalized Gini indices," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(3), pages 471-496, October.
    2. Francesco Andreoli & Claudio Zoli, 2015. "Measuring the interaction dimension of segregation: the Gini-Exposure index," Working Papers 30/2015, University of Verona, Department of Economics.
    3. Markus Jäntti & Eva M. Sierminska & Philippe Van Kerm, 2015. "Modeling the Joint Distribution of Income and Wealth," Research on Economic Inequality,in: Measurement of Poverty, Deprivation, and Economic Mobility, volume 23, pages 301-327 Emerald Publishing Ltd.
    4. Anderson, Gordon, 2011. "Polarization measurement and inference in many dimensions when subgroups can not be identified," Economics - The Open-Access, Open-Assessment E-Journal, Kiel Institute for the World Economy (IfW), vol. 5, pages 1-19.
    5. repec:hal:journl:halshs-00085881 is not listed on IDEAS
    6. Henar Diez & Mª Casilda Lasso de la Vega & Ana Marta Urrutia, 2007. "Unit-Consistent Aggregative Multidimensional Inequality Measures: A Characterization," Working Papers 66, ECINEQ, Society for the Study of Economic Inequality.
    7. Walter Krämer, 2016. "Walter Krämer: Interview mit Karl Mosler," AStA Wirtschafts- und Sozialstatistisches Archiv, Springer;Deutsche Statistische Gesellschaft - German Statistical Society, vol. 10(1), pages 63-71, February.
    8. K. Mosler, 2003. "Central regions and dependency," Econometrics 0309004, EconWPA.
    9. Chiara Gigliarano & Karl Mosler, 2009. "Constructing indices of multivariate polarization," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 7(4), pages 435-460, December.
    10. Karl Mosler, 2005. "Restricted Lorenz dominance of economic inequality in one and many dimensions," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 2(2), pages 89-103, January.
    11. John A. Weymark, 2003. "The Normative Approach to the Measurement of Multidimensional Inequality," Vanderbilt University Department of Economics Working Papers 0314, Vanderbilt University Department of Economics, revised Jan 2004.
    12. E. Abdul-Sathar & R. Suresh & K. Nair, 2007. "A vector valued bivariate gini index for truncated distributions," Statistical Papers, Springer, vol. 48(4), pages 543-557, October.
    13. Eisenberg, Bennett, 2015. "The multivariate Gini ratio," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 292-298.
    14. Masato Okamoto, 2009. "Decomposition of gini and multivariate gini indices," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 7(2), pages 153-177, June.
    15. Chiara GIGLIARANO & Karl MOSLER, 2009. "Measuring middle-class decline in one and many attributes," Working Papers 333, Universita' Politecnica delle Marche (I), Dipartimento di Scienze Economiche e Sociali.
    16. Gordon Anderson, 2008. "The empirical assessment of multidimensional welfare, inequality and poverty: Sample weighted multivariate generalizations of the Kolmogorov–Smirnov two sample tests for stochastic dominance," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 6(1), pages 73-87, March.
    17. Olena Nizalova, 2014. "Inequality in Total Returns to Work in Ukraine: Taking A Closer Look at Workplace (Dis)amenities," Discussion Papers 52, Kyiv School of Economics.
    18. Marco Dall’Aglio & Marco Scarsini, 2000. "Zonoids, Linear Dependence, and Size-Biased Distributions on the Simplex," ICER Working Papers - Applied Mathematics Series 27-2003, ICER - International Centre for Economic Research, revised Jul 2003.

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