On a new class of continuous indices of inequality

The Gini index is a well-known and long-established measure of inequality for distributions of income and other quantities. However, it has a number of mathematical disadvantages. Firstly, it is discontinuous with respect to all the main modes of convergence of probability measures. Secondly, it relies critically on the finiteness of the mean of the underlying distribution. Finally, even when the underlying distribution has a finite mean, estimation of the Gini index from data can be problematic if the variance of the underlying distribution is infinite. In this paper, we propose a class of inequality indices which are continuous with respect to setwise convergence of probability measures (and hence also with respect to convergence in total variation) and which do not require the underlying distribution to possess any finite moments whatsoever. Moreover, our class of inequality indices can be easily estimated from data and the standard methods of statistical inference can be applied to the estimators.