A Note on Finding Geodesic Equation of Two Parameters Gamma Distribution

William W. S. Chen

Department of Statistics, The George Washington University, Washington DC, USA.

DOI: 10.4236/am.2014.521328                

Abstract

Engineers commonly use the gamma distribution to describe the life span or metal fatigue of a manufactured item. In this paper, we focus on finding a geodesic equation of the two parameters gamma distribution. To find this equation, we applied both the well-known Darboux Theorem and a pair of differential equations taken from Struik [1]. The solution proposed in this note could be used as a general solution of the geodesic equation of gamma distribution. It would be interesting if we compare our results with Lauritzen’s [2].

Keywords

Darboux Theorem, Geodesic Equation, Differential Equation, Gamma Distribution

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Struik, D.J. (1961) Lectures on Classical Differential Geometry. 2nd Edition, Dover Publications, Inc., New York.
[2]	Lauritzen, S.L. (1987) Chapter 4: Statistical Manifolds. Differential Geometry in Statistical Inference. Vol. 10, Institute of Mathematical Statistics, Lecture Notes Monograph Series, Hayward, 163-216.
[3]	Rao, C.R. (1945) Information and the Accuracy Attainable in the Estimation of Statistical Parameters. Bulletin of Calcutta Mathematical Society, 37, 81-91.
[4]	Mitchell, A.F.S. (1988) Statistical Manifolds of Univariate Elliptic Distributions. International Statistical Review, 56, 1-16. http://dx.doi.org/10.2307/1403358
[5]	Oller, J.M. (1987) Information Metric for Extreme Value and Logistic Probability Distributions. Sankhya A, 49, 17-23.
[6]	Chen, W.W.S. (1998) Curvature: Gaussian or Riemann. International Conference (IISA), McMaster University, Hamilton, October.
[7]	Chen, W.W.S. and Kotz, S. (2013) The Riemannian Structure of the Three-Parameter Gamma Distribution. Applied Mathematics, 4, 514-522.
[8]	Darboux, G. (1889-1997) Lecons sur la theorie generale des surfaces. Gauthier-Villars, Paris.
[9]	Amari, S.I. (1982) Differential Geometry of Curved Exponential Families Curvature and Information Loss. Annals of Statistics, 10, 357-385. http://dx.doi.org/10.1214/aos/1176345779
[10]	Gray, A. (1993) Modern Differential Geometry of Curves and Surfaces. CRC Press, Inc., Boca Raton.