A distance between elliptical distributions based in an embedding into the Siegel group

This paper describes two different embeddings of the manifolds corresponding to many elliptical probability distributions with the informative geometry into the manifold of positive-definite matrices with the Siegel metric, generalizing a result published previously elsewhere. These new general embeddings are applicable to a wide class of elliptical probability distributions, in which the normal, t-Student and Cauchy are specific examples. A lower bound for the Rao distance is obtained, which is itself a distance, and, through these embeddings, a number of statistical tests of hypothesis are derived.