On Elliptical and Inverse Elliptical Wishart distributions

Abstract: This paper deals with the Elliptical Wishart and Inverse Elliptical Wishart distributions, which play a major role when handling covariance matrices. Similarly to multivariate elliptical distributions, these form a large family of covariance distributions, encompassing, e.g., the Wishart or \textit{t}-Wishart ones. Our first major contribution is to derive a stochastic representation for Elliptical Wishart and Inverse Elliptical Wishart matrices. This later enables us to obtain various key statistical properties of Elliptical Wishart and Inverse Elliptical Wishart distributions such as expectations, variances, and Kronecker moments up to any orders. The stochastic representation also allows us to provide an efficient method to generate random matrices from Elliptical Wishart and Inverse Elliptical Wishart distributions. Finally, the practical interest of Elliptical Wishart distributions - in particular the \textit{t}-Wishart one - is demonstrated through a fitting experiment on real electroencephalographic data. This showcases their effectiveness in accurately modeling real covariance matrices.