Stochastic orderings of multivariate elliptical distributions

Let ${\bf X}$ and 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 provide a unified derivation of sufficient necessary conditions classifying multivariate vectors according to several main integral stochastic orders. As consequence obtain new inequalities by applying it distributions. The results generalize the corresponding ones normal in literature.