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Multivariate Gini Indices

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  • Koshevoy, G. A.
  • Mosler, K.

Abstract

Two extensions of the univariate Gini index are considered:RD, based on expected distance between two independent vectors from the same distribution with finite mean[mu][set membership, variant]d; andRV, related to the expected volume of the simplex formed fromd+1 independent such vectors. A new characterization ofRDas proportional to a univariate Gini index for a particular linear combination of attributes relates it to the Lorenz zonoid. TheLorenz zonoidwas suggested as a multivariate generalization of the Lorenz curve.RVis, up to scaling, the volume of the Lorenz zonoid plus a unit cube of full dimension. Whend=1, bothRDandRVequal twice the area between the usual Lorenz curve and the line of zero disparity. Whend>1, they are different, but inherit properties of the univariate Gini index and are related via the Lorenz zonoid:RDis proportional to the average of the areas of some two-dimensioned projections of the lift zonoid, whileRVis the average of the volumes of projections of the Lorenz zonoid over all coordinate subspaces.

Suggested Citation

  • Koshevoy, G. A. & Mosler, K., 1997. "Multivariate Gini Indices," Journal of Multivariate Analysis, Elsevier, vol. 60(2), pages 252-276, February.
  • Handle: RePEc:eee:jmvana:v:60:y:1997:i:2:p:252-276
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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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