Base Change for Fisher Metrics: Case of the q−Gaussian Inverse Distribution
It is known that the Riemannian manifold determined by the family of inverse Gaussian distributions endowed with the Fisher metric has negative constant curvature κ = −1/2 , as does the family of usual Gaussian distributions. In the present paper, firstly we arrive at this result by following a different path, much simpler than the previous ones. We first put the family in exponential form, thus endowing the family with a new set of parameters, or coordinates, θ1, θ2; then we determine the matrix of the Fisher metric in terms of these parameters; and finally we compute this matrix in the original parameters. Secondly, we define the Inverse q−Gaussian distribution family (q < 3), as the family obtained by replacing the usual exponential function by the Tsallis q−exponential function in the expression for the Inverse Gaussian distribution, and observe that it supports two possible geometries, the Fisher and the q−Fisher geometry. And finally, we apply our strategy to obtain results about the Fisher and q−Fisher geometry of the Inverse q−Gaussian distribution family, similar to the ones obtained in the case of the Inverse Gaussian distribution family.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 231
 Amari, Shun-ichi and Ohara, Atsumi. Geometry of q-Exponential Family of Probability Distributions. Entropy 13, 2011, pp. 1170-1185. DOI: 10.3390/e13061170.
 Amari, Shun-ichi and Nagaoka, Hiroshi. Methods of Information Geometry. Translation of Mathematics Monographs, volume 191. American Mathematics Society, Oxford University Press, 2000.
 Cadavid, Carlos; Loaiza, Gabriel, and Arango, Juan Carlos. Geodesic in the manifold generated by the q-Gaussian distribution. Differential Geometry and its Applications. It is currently under review, submitted since April in 2021.
 Chhikara, Raj and Folks, Leroy. The inverse Gaussian distribution: theory, methodology and applications. Marcel Dekker, Inc., New York, 1989.
 Dodson, C.T.J. Information Geometry. School Mathematics, Manchester University. DOI:10.1109/TMI.2010.2086464.
 Ghoshdastidar, Debarghya. On some Statistical Properties of Multivariate q-Gaussian Distribution and its application to Smoothed Functional Algorithms. International Symposium on Information Theory. IEEE, 2012.
 Loaiza; Gabriel and Quiceno; H´ector. A q-exponential statistical Banach manifold. Journal of Mathematical Analysis and Applications, 398, 2013. DOI: https://doi.org/10.1016/j.jmaa.2012.08.046.
 Matsuzoe, Hiroshi and Ohara, Atsumi. Geometry for a q-exponential families. WSPC - Proceedings, April 16, 2011. DOI: https://doi.org/10.1142/9789814355476 0004.
 Naudts, Jan. Generalised Thermostatistics. Springer, London, 2011. DOI: https://doi.org/10.1007/978-0-85729-355-8.
 Naudts, Jan. The q-exponential family in statistical physics. Central European Journal of Physics September 2009, Volume 7, Issue 3, pp 405-413. DOI: 10.1088/1742-6596/201/1/012003.
 Tanaya, Daiki, Tanaka, Masaru and Matsuzoe, Hiroshi. Notes on geometry of q-normal distributions. WSPC - Proceedings, May 07, 2011. DOI: https://doi.org/10.1142/9789814355476 0009.
 Tsallis, Constantino. Introduction to Nonextensive Statistical Mechanics. Approaching a complex world. Springer, 2009. DOI 10.1007/978-0-387-85359-8.
 Wani, J. K. and Kare, D. G. Note on a Characterization of the Inverse Gaussian Distribution. The Annals of Mathematical Statistics, 1970, Vol. 41, No. 3.
 Zhang, Zhenning; Sun, Huafei and Zhong, Fengwei. Information geometry power inverse Gaussian distribution. Applied Sciences, Vol.9, 2007, pp. 194-203.