Preferences help
enabled [disable] Abstract
Number of results
2020 | 143 | 181-202
Article title

The Geometrical Structures of Bivariate Gamma Exponential Distributions

Title variants
Languages of publication
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.
Physical description
  • Mathematics Department, Tripoli University, Tripoli, Libya
  • [1] N. H. Abdel-All, M. A. W. Mahmoud and H. N. Abd-Ellah. Geomerical Properties of Pareto Distribution. Applied Mathematics and Computation 145 (2003) 321-339
  • [2] Amari S. and Nagaoka H. Methods of Information Geometry. American Mathematical Society, Oxford University Press (2000).
  • [3] Khadiga Arwini, C.T.J. Dodson. Information Geometry Near Randomness and Near Independence. Lecture Notes in Mathematics 1953, Springer-Verlag, Berlin, Heidelberg, New York (2008).
  • [4] Khadiga Arwini, C.T.J. Dodson. Information Geometric Neighbourhoods of Randomness and Geometry of the Mckay Bivariate Gamma 3-manifold. Sankhya: Indian Journal of Statistics 66 (2) (2004) 211-231
  • [5] C.T.J. Dodson. Systems of connections for parametric models. In Proc. Workshop on Geometrization of Statistical Theory (1987) 28-31.
  • [6] B. Efron. Defining the curvature of a statistical problem (with application to second order efficiency) (with discussion). Ann. Statist. 3 (1975) 1189-1242
  • [7] Samuel Kotz, N. Balakrishnan and Norman L. Johnson. Continuous Multivariate distributions. Volume 1: Models and Applications, Second Edition, John Wiley and Sons, New York (2000).
  • [8] Saralees Nadarajah. The Bivariate Gamma Exponential Distribution With Application To Drought Data. Journal of applied Mathematics and computing, 24 (2007) 221-230
  • [9] Jose M. Oller. Information metric for extreme value and logistic probability distributions. Sankhya: The Indian Journal of Statistics, 49 (1987), Series A, Pt. 1, pp. 17-23
  • [10] C.R. Rao. Information and accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37 (1945) 81-91
  • [11] K. Nomizu and T. Sasaki, Affine differential geometry: Geometry of Affine Immersions. Cambridge University Press, Cambridge (1994).
  • [12] Y. Sato, K. Sugawa and M. Kawaguchi. The geometrical structure of the parameter space of the two-dimensional normal distribution. Division of information engineering, Hokkaido University, Sapporo, Japan (1977).
  • [13] L.T. Skovgaard. A Riemannian geometry of the multivariate normal model. Scand. J. Statist. 11 (1984) 211-223
  • [14] Steel, S. J., and le Roux, N. J. A class of compound distributions of the reparametrized bivariate gamma extension. South African Statistical Journal 23 (1989) 131-141
  • [15] S. Yue. A Bivariate Gamma Distribution for Use in Multivariate Flood Frequency Analysis. Hydrological Processes 15 (2001) 1033-1045
  • [16] S. Yue. Applying bivariate normal distribution to flood frequency analysis. Water International 24 (1999) 248-254
  • [17] S. Yue, Applicability of the Nagao–Kadoya bivariate exponential distribution for modeling two correlated exponentially distributed variates. Stochastic Environmental Research and Risk Assessment 15 (2001) 244-260
  • [18] S. Yue, T. B. M. J. Ouarda and B. Bobee. A Review of Bivariate Gamma Distributions for Hydrological Application. Journal of Hydrology 246 (2001) 1-18
Document Type
Publication order reference
YADDA identifier
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.