Ordering Gini Indexes of Multivariate Elliptical Risks

Abstract

In this paper, we establish several stochastic orders between Gini indexes of multivariate elliptical risks with the same marginals but different dependence structures. This work is motivated by the studies of Brazauskas et al (2007) and Samanthi et al (2015), who employed the Gini index to compare the riskiness of insurance portfolios. Using Monto Carlo simulations, they found that that the power function and probability of type I error of the associated hypothesis test increase as portfolios become more positively correlated. The comparison of Gini indexes presented in this paper provides a theoretical explanation to this statistical phenomenon. Moreover, it enriches the studies of the problem of central concentration of multivariate elliptical distributions and generalizes the pd-1 order proposed by Shaked and Tong (1985).

Keywords: Gini index, elliptical distribution, dependence structure, comonotonicity, usual stochastic order, increasing convex order, supermodular order, pd-1 order