A Note on Finding Geodesic Equation of Two Parameters Gamma Distribution

Author(s)
William W. S. Chen

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].

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].

Cite this paper

Chen, W. (2014) A Note on Finding Geodesic Equation of Two Parameters Gamma Distribution.*Applied Mathematics*, **5**, 3511-3517. doi: 10.4236/am.2014.521328.

Chen, W. (2014) A Note on Finding Geodesic Equation of Two Parameters Gamma Distribution.

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.

[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.