An Identity for Expectations and Characteristic Function of Matrix Variate Skew-normal Distribution with Applications to Associated Stochastic Orderings

We establish an identity for EfY-EfX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ef\left( \varvec{Y}\right) - Ef\left( \varvec{X}\right) $$\end{document}, when X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{X}$$\end{document} and Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{Y}$$\end{document} both have matrix variate skew-normal distributions and the function f satisfies some weak conditions. The characteristic function of matrix variate skew normal distribution is then derived. We then make use of it to derive some necessary and sufficient conditions for the comparison of matrix variate skew-normal distributions under six different orders, such as usual stochastic order, convex order, increasing convex order, upper orthant order, directionally convex order and supermodular order.