Stochastic ordering of Gini indexes for multivariate elliptical risks

Abstract

In this paper, we show that the conjecture, made by Samanthi et al. (2016), on the ordering of Gini indexes of multivariate normal risks with respect to the strength of dependence, is not true. By using the positive semi-definite ordering of covariance matrices, we can obtain the usual stochastic order of the Gini indexes for multivariate normal risks. This can be generalized to multivariate elliptical risks. We also investigate the monotonicity of the Gini indexes in the usual stochastic order when the covariance (dispersion, resp.) matrices of multivariate normal (elliptical, resp) risks increase componentwise. In addition, we derive a large deviation result for the Gini indexes of multivariate normal risks.