The Fisher–Rao Geometry of CES Distributions

When dealing with a parametric statistical model, a Riemannian manifold can naturally appear by endowing the parameter space with the Fisher information metric. The geometry induced on the parameters by this metric is then referred to as the Fisher–Rao information geometry. Interestingly, this yields a point of view that allows for leveraging many tools from differential geometry. After a brief introduction about these concepts, we will present some practical uses of these geometric tools in the framework of elliptical distributions. This second part of the exposition is divided into three main axes: Riemannian optimization for covariance matrix estimation, Intrinsic Cramér–Rao bounds, and classification using Riemannian distances.