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Multivariate Gini indices. (English) Zbl 0873.62062
Summary: 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 \(\mu\in \mathbb{R}^d\); 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.

MSC:
62H99 Multivariate analysis
60E15 Inequalities; stochastic orderings
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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