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Multivariate Gini Indices

  • Koshevoy, G. A.
  • Mosler, K.

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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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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