Discussion Papers in Statistics and Econometrics Seminar of Economic and Social Statistics University of Cologne

The Gini index and the Gini mean di erence of a univariate distribution are extended to measure the disparity of a general d-variate distribution. We propose and investigate two approaches, one based on the distance of the distribution from itself, the other on the volume of a convex set in (d + 1)space, named the lift zonoid of the distribution. When d = 1, this volume equals the area between the usual Lorenz curve and the line of zero disparity, up to a scale factor. We get two de nitions of the multivariate Gini index, which are di erent (when d > 1) but connected through the notion of the lift zonoid. Both notions inherit properties of the univariate Gini index, in particular, they are vector scale invariant, continuous, bounded by 0 and 1, and the bounds are sharp. They vanish if and only if the distribution is concentrated at one point. The indices have a ceteris paribus property and are consistent with multivariate extensions of the Lorenz order. Illustrations with data conclude the paper.