# zbMATH — the first resource for mathematics

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
Full Text: