Information Geometry of Frechet Distributions
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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.
-  C.R. Rao, Information and accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37 (1945) 81-91
-  F B. Efron, Defining the curvature of a statistical problem (with application to second order efficiency) (with discussion). Ann. Statist 3 (1975) 1189-1242
-  Amari S. and Nagaoka H, Methods of Information Geometry. American Mathematical Society, Oxford University Press (2000).
-  C.T.J. Dodson, Systems of connections for parametric models. In Proc. Workshop on Geometrization of Statistical Theory. Lancaster, October 28-31 (C. T. J. Dodson, eds.), ULDS Publications, Univ. Lancaster, 1987.
-  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).
-  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
-  Khadiga Arwini and C.T.J. Dodson, Neighbourhoods of independence and associated geometry. Central European Journal of Mathematics 5 (1) (2007) 50-83
-  Khadiga Arwini, C.T.J. Dodson and Hiroshi Matsuzoe, Alpha connections and affine embedding of McKay bivariate gamma 3-manifold. International. J. Pure Appl. Math. 9 (2) (2003) 253-262
-  Khadiga Arwini, L. Del Riego and C.T.J. Dodson, Universal connection and curvature for statistical manifold geometry. Houston Journal of Mathematics 33 (1) (2007) 145-161
-  N. H. Abdel-All, M. A. W. Mahmoud and H. N. Abd-Ellah, Geometrical Properties of Pareto Distribution. Applied Mathematics and Computation 145 (2003) 321-339
-  Limei Cao, Huafei Sun and Xiaojie Wang, The geometrical structures of the Weibull distribution manifold and the generalized exponential distribution manifold. Tamkang Journal of Mathematics 39 (1) (2008) 45-51
-  Jose M. Oller, Information metric for extreme value and logistic probability distributions sankhya. The Indian Journal of Statistics, Vol 49, A (1) (1987) 17-23
-  Frechet, M. Sur la loi de probabilite de l’ecart maximum Ann. Soc. Polon. Math. 6 (93) (1927).
-  Pedro L. Ramos, Francisco Louzada, Eduardo Ramos and Sanku Dey, The Fréchet distribution: Estimation and application - An overview. Journal of Statistics and Management Systems (2019). https://doi.org/10.1080/09720510.2019.1645400
-  W. Nasir and M. Aslam, Bayes approach to stud shape parameter of Frechet distribution. International Journal of Basic and Applied Science 4 (3) (2015) 246-254
-  Harlow D. G, Application of the Frechet distribution function. Int. J. of Mat. & Prod. Tech (17) (2002) 482-495
-  Mubarak M, (20120). Parameter estimation based on the Frechet progressive type II censored data with binomial removals. International Journal of Quality, Statistics, and Reliability Volume 2012, Article ID 245910, 5 pages. doi:10.1155/2012/245910
-  Kamran Abbas, Nosheen Yousaf Abbasi, Amjad Ali and Sajjad Ahmad Khan, (2018). Bayesian analysis of three-parameters Frechet Distribution with medical applications. Computational and Mathematical Methods in Medicine Volume 2019, Article ID 9089856, 8 page. shttps://doi.org/10.1155/2019/9089856
-  Nadarajah. S and Kotz. S, Sociological models based on Frechet random variables. Quality and Quantity 42 (2008) 89-85
-  M. Mubarak, Estimation of the Frechet distribution parameters on the record values. Arabian Journal of Science and Engineering 36 (8) (2011) 1597-1606
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