Stochastic Orderings of Multivariate Elliptical Distributions

Let $\mathbf{X}$ and $\mathbf{X}$ be two $n$-dimensional elliptical random vectors, we establish an identity for $E[f(\mathbf{Y})]-E[f(\mathbf{X})]$, 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.