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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.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Multivariate Analysis.

Volume (Year): 60 (1997)
Issue (Month): 2 (February)
Pages: 252-276

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Handle: RePEc:eee:jmvana:v:60:y:1997:i:2:p:252-276

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Related research

Keywords: dilation disparity measurement Gini mean difference lift zonoid Lorenz order;

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Cited by:
  1. Thibault Gajdos & John Weymark, 2005. "Multidimensional Generalized Gini Indices," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00085881, HAL.
  2. 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.
  3. 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.
  4. Karl Mosler, 2005. "Restricted Lorenz dominance of economic inequality in one and many dimensions," Journal of Economic Inequality, Springer, vol. 2(2), pages 89-103, January.
  5. 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.
  6. 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.
  7. 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.
  8. Thibault Gajdos & John Weymark, 2005. "Multidimensional Generalized Gini Indices," Post-Print halshs-00085881, HAL.
  9. Masato Okamoto, 2009. "Decomposition of gini and multivariate gini indices," Journal of Economic Inequality, Springer, vol. 7(2), pages 153-177, June.
  10. 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, vol. 6(1), pages 73-87, March.
  11. Chiara Gigliarano & Karl Mosler, 2009. "Constructing indices of multivariate polarization," Journal of Economic Inequality, Springer, vol. 7(4), pages 435-460, December.
  12. K. Mosler, 2003. "Central regions and dependency," Econometrics 0309004, EconWPA.
  13. 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.

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