Inequality measures for multivariate distributions

Abstract

The Lorenz order and related summary measures of inequality, such as the celebrated Gini Index, seem to be quite generally accepted as appropriate indicators of inequality in univariate populations. In multivariate settings, the issue of how one should measure and compare inequality in populations does not appear to be as clearly resolved. Several extensions of the Lorenz order and the Gini index have been proposed. At present the Lorenz zonoid ordering appears to be the generalization of the univariate Lorenz order that is most likely to command general acceptance. The issue is not as clear with regard to multivariate inequality measures. Several candidate measures will be described and their properties discussed. In the final analysis, the volume of the Lorenz zonoid appears to be a strong candidate for the title of "natural extension of the Gini index to higher dimensions".