Stochastic Orderings of Multivariate Elliptical Distributions

TL;DR

This paper develops a unified framework for comparing multivariate elliptical distributions using stochastic orders, generalizing results from the normal case and deriving new inequalities.

Contribution

It introduces an identity for expectations of functions of elliptical vectors, enabling a unified approach to stochastic ordering and inequality derivation.

Findings

Provides necessary and sufficient conditions for stochastic orders among elliptical vectors.

Derives new inequalities for multivariate elliptical distributions.

Generalizes known results from the multivariate normal case.

Abstract

Let ${\bf X}$ and ${\bf X}$ be two $n$-dimensional elliptical random vectors, we establish an identity for $E[f({\bf Y})]-E[f({\bf X})]$, where $f: \Bbb{R}^n \rightarrow \Bbb{R}$ fulfilling some regularity conditions. Using this identity we provide a unified derivation of sufficient and necessary conditions for classifying multivariate elliptical random vectors according to several main integral stochastic orders. As a consequence we obtain new inequalities by applying it to multivariate elliptical distributions. The results generalize the corresponding ones for multivariate normal random vectors in the literature.