Information Geometry of Frechet Distributions

Using the Fisher information matrix (FIM) as a Riemannian metric, the family of Frechet distributions determines a two dimensional Riemannian manifold. In this paper we illustrates the information geometry of the Frechet space, and derive the α-geometry as; α-connections, α-curvature tensor, α-Ricci curvature with its eigenvalues and eigenvectors, α-sectional curvature, α-mean curvature, and α-scalar curvature, where we show that Frechet space has a constant α-scalar curvature. The special case where α = 0 corresponds to the geometry induced by the Levi-Civita connection. In addition, we consider three special cases of Frechet distributions as submanifolds with dimension one, and discuss their geometrical structures, then we prove that one of these submanifolds is an isometric isomorph of the exponential manifold, which is important in stochastic process since exponential distributions represent intervals between events for Poisson processes. After that, we introduce log-Frechet distributions, and show that this family of distributions determines a Riemannian 2-manifold which is isometric with the origin manifold. Finally, an explicit expressions for some distances in Frechet space are obtained as, Kullback-Leibler distance, and J-divergence.