Multivariate Gini indices.

*(English)*Zbl 0873.62062Summary: 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 |