Multivariate gini indices

[摘要] Two extensions of the univariate Gini index are considered: R(D), based on expected distance between two independent vectors from the same distribution with finite mean mu is an element of R(d); and R(v), related to the expected volume of the simplex formed From d + 1 independent such vectors. A new characterization of R(D) as proportional to a univariate Gini index for a particular linear combination of attributes relates it to the Lorenz zonoid. The Lorenz zonoid was suggested as a multivariate generalization of the Lorenz curve. R(v) is, up to scaling, the volume of the Lorenz zonoid plus a unit cube of full dimension. When d = 1, both R(D) and R(v) equal twice the area between the usual Lorenz curve and the line of zero disparity. When d > 1, they are different, but inherit properties of the univariate Gini index and are related via the Lorenz zonoid: R(D) is proportional to the average of the areas of some two-dimensioned projections of the lift zonoid, while R(v) is the average of the volumes of projections of the Lorenz zonoid over all coordinate subspaces. (C) 1997 Academic Press.

[发布日期] 1997-02-01

[关键词] dilation;disparity measurement;Gini mean difference;lift zonoid;Lorenz order