On Generalized Wishart Distributions - I: Likelihood Ratio Test for Homogeneity of Covariance Matrices

In this paper, we define and discuss a class of generalized Wishart distributions under elliptical models. We derive the non-central moments of the likelihood ratio statistic for testing the equality of two covariance matrices under elliptical models for the corresponding matrices. Known classical expressions for the Gaussian model are then deduced from these general results. Finally, the exact distribution of the Wilks’ statistic under a specific distribution, including the Gaussian distribution as a particular member, is derived.