# Stochastic comparisons of sample mean differences for multivariate random variables

## TL;DR

This paper extends stochastic ordering results of the Gini index from multivariate normal risks to more general multivariate elliptical risks, analyzing their dispersion matrices and tail probabilities.

## Contribution

It generalizes existing stochastic ordering results for Gini indexes to multivariate elliptical risks and revises large deviation results for these risks.

## Key findings

- Conditions on dispersion matrices ensure monotonicity of Gini index
- Established stochastic ordering for multivariate elliptical risks
- Revised large deviation results for Gini indexes

## Abstract

In this paper, we establish the stochastic ordering of the Gini indexes for multivariate elliptical risks which generalized the corresponding results for multivariate normal risks. It is shown that several conditions on dispersion matrices and the components of dispersion matrices of multivariate normal risks for the monotonicity of the Gini index in the usual stochastic order proposed by Samanthi, Wei and Brazauskas (2016) and Kim and Kim (2019) are also suitable for multivariate elliptical risks. We also study the tail probability of Gini index for multivariate elliptical risks and revised a large deviation result for the Gini indexes of multivariate normal risks in Kim and Kim (2019).