The Geometrical Structures of Bivariate Gamma Exponential Distributions

This paper is devoted to the information geometry of the family of bivariate gamma exponential distributions, which have gamma and Pareto marginals, and discuss some of its applications. We begin by considering the parameter bivariate gamma exponential manifold as a Riemannian 3-manifold; by following Rao’s idea to use the Fisher information matrix (FIM), and derive the α-geometry as: α-connections, α-curvature tensor, α-Ricci curvature with its eigenvalues and eigenvectors, and α-scalar curvature. Where here the 0-geometry corresponds to the geometry induced by the Levi-Civita connection, and we show that this space has a non-constant negative scalar curvature. In addition, we consider four submanifolds as special cases, and discuss their geometrical structures, and we prove that one of these submanifolds is an isometric isomorph of the univariate gamma manifold. Then we introduce log-bivariate gamma exponential distributions, which have log-gamma and log-Pareto marginals, and we show that this family of distributions determines a Riemannian 3-manifold which is isometric with the origin manifold. We give an analytical solution for the geodesic equations, and obtain the explicit expressions for Kullback-Leibler distance, J-divergence and Bhattacharyya distance. Finally, we prove that the bivariate gamma exponential manifold can be realized in R4, using information theoretic immersions, and we give explicit information geometric tubular neighbourhoods for some special cases.