The Fisher-Rao geometry of CES distributions

TL;DR

This paper explores the Fisher-Rao geometry of CES distributions, applying differential geometric tools to improve covariance estimation, derive bounds, and enhance classification in elliptical distribution models.

Contribution

It introduces the application of Fisher-Rao information geometry to CES distributions, enabling advanced Riemannian optimization, bounds, and classification methods.

Findings

Enhanced covariance matrix estimation via Riemannian optimization

Derived intrinsic Cramér-Rao bounds for elliptical distributions

Improved classification accuracy using Riemannian distances

Abstract

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 leveragingmany 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 Cram\'er-Rao bounds, and classification using Riemannian distances.