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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. --

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Bibliographic Info

Paper provided by University of Cologne, Department for Economic and Social Statistics in its series Discussion Papers in Statistics and Econometrics with number 7/95.

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Date of creation: 1995
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Handle: RePEc:zbw:ucdpse:9507

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Keywords: Dilation; Disparity measurement; Gini mean difference; Lift zonoid; Lorenz order;

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Cited by:
  1. Thibault Gajdos & John A. Weymark, 2003. "Multidimensional Generalized Gini Indices," Vanderbilt University Department of Economics Working Papers 0311, Vanderbilt University Department of Economics, revised Jul 2003.
  2. Karl Mosler, 2004. "Restricted Lorenz dominance of economic inequality in one and many dimensions," Journal of Economic Inequality, Springer, Springer, vol. 2(2), pages 89-103, August.
  3. 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.
  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, vol. 5(11), pages 1-19.
  5. Nizalova, Olena Y., 2014. "Inequality in Total Returns to Work in Ukraine: Taking a Closer Look at Workplace (Dis)amenities," IZA Discussion Papers 8322, Institute for the Study of Labor (IZA).
  6. K. Mosler, 2003. "Central regions and dependency," Econometrics, EconWPA 0309004, EconWPA.
  7. repec:hal:journl:halshs-00085881 is not listed on IDEAS
  8. Chiara GIGLIARANO & Karl MOSLER, 2009. "Measuring middle-class decline in one and many attributes," Working Papers, Universita' Politecnica delle Marche (I), Dipartimento di Scienze Economiche e Sociali 333, Universita' Politecnica delle Marche (I), Dipartimento di Scienze Economiche e Sociali.
  9. E. Abdul-Sathar & R. Suresh & K. Nair, 2007. "A vector valued bivariate gini index for truncated distributions," Statistical Papers, Springer, Springer, vol. 48(4), pages 543-557, October.
  10. Chiara Gigliarano & Karl Mosler, 2009. "Constructing indices of multivariate polarization," Journal of Economic Inequality, Springer, Springer, vol. 7(4), pages 435-460, December.
  11. 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," Journal of Economic Inequality, Springer, Springer, vol. 6(1), pages 73-87, March.
  12. 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.
  13. Marco Dall’Aglio & Marco Scarsini, 2000. "Zonoids, Linear Dependence, and Size-Biased Distributions on the Simplex," ICER Working Papers - Applied Mathematics Series, ICER - International Centre for Economic Research 27-2003, ICER - International Centre for Economic Research, revised Jul 2003.
  14. 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.
  15. Masato Okamoto, 2009. "Decomposition of gini and multivariate gini indices," Journal of Economic Inequality, Springer, Springer, vol. 7(2), pages 153-177, June.

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