The Riemannian Structure of the Three-Parameter Gamma Distribution

ABSTRACT

In this paper, we will utilize the results already known in differential geometry and provide an intuitive understanding of the Gamma Distribution. This approach leads to the definition of new concepts to provide new results of statistical importance. These new results could explain Chen [1-3] experienced difficulty when he attempts to simulate the sampling distribution and power function of Cox’s [4,5] test statistics of separate families of hypotheses. It may also help simplify and clarify some known statistical proofs or results. These results may be of particular interest to mathematical physicists. In general, it has been shown that the parameter space is not of constant curvature. In addition, we calculated some invariant quantities, such as Sectional curvature, Ricci curvature, mean curvature and scalar curvature.

In this paper, we will utilize the results already known in differential geometry and provide an intuitive understanding of the Gamma Distribution. This approach leads to the definition of new concepts to provide new results of statistical importance. These new results could explain Chen [1-3] experienced difficulty when he attempts to simulate the sampling distribution and power function of Cox’s [4,5] test statistics of separate families of hypotheses. It may also help simplify and clarify some known statistical proofs or results. These results may be of particular interest to mathematical physicists. In general, it has been shown that the parameter space is not of constant curvature. In addition, we calculated some invariant quantities, such as Sectional curvature, Ricci curvature, mean curvature and scalar curvature.

Cite this paper

W. Chen and S. Kotz, "The Riemannian Structure of the Three-Parameter Gamma Distribution,"*Applied Mathematics*, Vol. 4 No. 3, 2013, pp. 514-522. doi: 10.4236/am.2013.43077.

W. Chen and S. Kotz, "The Riemannian Structure of the Three-Parameter Gamma Distribution,"

References

[1] W.W. S. Chen, “Testing Gamma and Weibull Distribution: A Comparative Study,” Estadistica, Vol. 39, 1987, pp. 1-26.

[2] W. W. S. Chen, “Evaluation of the First 12 Derivatives of the Digamma PSI Functions with Applications,” Proceeding of Statistical Computing Section, 1982, pp. 293-298.

[3] W. W. S. Chen, “Curvature Gaussian or Riemann,” International Conference (IISA), McMaster University, Hamilton, 10-11 October 1998.

[4] D. R. Cox, “Tests of Separate Families of Hypotheses,” Proceedings of 4th Berkeley Symposium, Vol. 1, 1961, pp. 105-123.

[5] D. R Cox, “Further Results on Tests of Separate Families of Hypotheses,” Society B, Vol. 24, 1962, pp. 406-424.

[6] C. R. Rao, “Information and Accuracy Attainable in the Estimation of Statistical Parameters,” Bulletin of Calcutta Mathematical Society, Vol. 37, 1945, pp. 81-89.

[7] B. Efron, “Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency),” Annals of Statistics, Vol. 3, No. 6, 1975, pp. 1189-1217. doi:10.1214/aos/1176343282

[8] A. F. S. Mitchell, “Statistical Manifolds of Univariate Elliptic Distributions,” International Statistical Review, Vol. 56, No. 1, 1988, pp. 1-16. doi:10.2307/1403358

[9] J. Burbea and C. R. Rao, “Entropy Differential Metric, Distance and Divergence Measures in Probability Spaces: A Unified Approach,” Journal Multivariate Analysis, Vol. 12, No. 4, 1982, pp. 575-596. doi:10.1016/0047-259X(82)90065-3

[10] L. T. Skovgaard, “A Riemannian Geometry of the Multivariate Normal Model,” Scandinavian Journal of Statistics, Vol. 11, No. 4, 1984, pp. 211-223.

[11] Y. Sato, K. Sugawa and M. Kawaguchi, “The Geometrical Structure of the Parameter Space of the Two-Dimensional Normal Distribution,” Reports on Mathematical Physics, Vol. 16, No. 1, 1979, pp. 111-119. doi:10.1016/0034-4877(79)90043-0

[12] R. E. Kass, “The Geometry of Asymptotic Inference,” Statistical Science, Vol. 4, No. 3, 1989, pp. 188-234.

[13] R. E. Kass and P. W. Vos, “Geometrical Foundations of Asymptotic Inference,” John Wiley & Sons, Inc., New York, 1997.

[14] A. C. Hearn, “REDUCE User’s and Contributed Packages Manual,” Version 3.7.

[15] N. Balakrishnan and W. Chen, “Handbook of Tables for Order Statistics from Gamma Distributions with Applications,” Kluwer Academic Publishers, Not Publish yet.

[16] N. L. Johnson and S. Kotz, “Continuous Univariate Distributions-1,” Houghton Mifflin Company, Boston, 1970.

[17] N. L. Johnson, S. Kotz and N. Balakrishnan, “Continuous Univariate Distributions,” 2nd Edition, John Wiley & Sons, Inc., New York, 1994.

[18] D. J. Struik, “Lectures on Classical Differential Geometry,” 2nd Edition, Dover Publications, Inc., New York, 1998.

[19] S. I. Goldberg, “Curvature and Homology,” Revised Edition, Dover Publications, Inc., New York, 1998.

[20] E. Kreyszig, “Differential Geometry,” Dover Publications, Inc., New York, 1991.

[1] W.W. S. Chen, “Testing Gamma and Weibull Distribution: A Comparative Study,” Estadistica, Vol. 39, 1987, pp. 1-26.

[2] W. W. S. Chen, “Evaluation of the First 12 Derivatives of the Digamma PSI Functions with Applications,” Proceeding of Statistical Computing Section, 1982, pp. 293-298.

[3] W. W. S. Chen, “Curvature Gaussian or Riemann,” International Conference (IISA), McMaster University, Hamilton, 10-11 October 1998.

[4] D. R. Cox, “Tests of Separate Families of Hypotheses,” Proceedings of 4th Berkeley Symposium, Vol. 1, 1961, pp. 105-123.

[5] D. R Cox, “Further Results on Tests of Separate Families of Hypotheses,” Society B, Vol. 24, 1962, pp. 406-424.

[6] C. R. Rao, “Information and Accuracy Attainable in the Estimation of Statistical Parameters,” Bulletin of Calcutta Mathematical Society, Vol. 37, 1945, pp. 81-89.

[7] B. Efron, “Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency),” Annals of Statistics, Vol. 3, No. 6, 1975, pp. 1189-1217. doi:10.1214/aos/1176343282

[8] A. F. S. Mitchell, “Statistical Manifolds of Univariate Elliptic Distributions,” International Statistical Review, Vol. 56, No. 1, 1988, pp. 1-16. doi:10.2307/1403358

[9] J. Burbea and C. R. Rao, “Entropy Differential Metric, Distance and Divergence Measures in Probability Spaces: A Unified Approach,” Journal Multivariate Analysis, Vol. 12, No. 4, 1982, pp. 575-596. doi:10.1016/0047-259X(82)90065-3

[10] L. T. Skovgaard, “A Riemannian Geometry of the Multivariate Normal Model,” Scandinavian Journal of Statistics, Vol. 11, No. 4, 1984, pp. 211-223.

[11] Y. Sato, K. Sugawa and M. Kawaguchi, “The Geometrical Structure of the Parameter Space of the Two-Dimensional Normal Distribution,” Reports on Mathematical Physics, Vol. 16, No. 1, 1979, pp. 111-119. doi:10.1016/0034-4877(79)90043-0

[12] R. E. Kass, “The Geometry of Asymptotic Inference,” Statistical Science, Vol. 4, No. 3, 1989, pp. 188-234.

[13] R. E. Kass and P. W. Vos, “Geometrical Foundations of Asymptotic Inference,” John Wiley & Sons, Inc., New York, 1997.

[14] A. C. Hearn, “REDUCE User’s and Contributed Packages Manual,” Version 3.7.

[15] N. Balakrishnan and W. Chen, “Handbook of Tables for Order Statistics from Gamma Distributions with Applications,” Kluwer Academic Publishers, Not Publish yet.

[16] N. L. Johnson and S. Kotz, “Continuous Univariate Distributions-1,” Houghton Mifflin Company, Boston, 1970.

[17] N. L. Johnson, S. Kotz and N. Balakrishnan, “Continuous Univariate Distributions,” 2nd Edition, John Wiley & Sons, Inc., New York, 1994.

[18] D. J. Struik, “Lectures on Classical Differential Geometry,” 2nd Edition, Dover Publications, Inc., New York, 1998.

[19] S. I. Goldberg, “Curvature and Homology,” Revised Edition, Dover Publications, Inc., New York, 1998.

[20] E. Kreyszig, “Differential Geometry,” Dover Publications, Inc., New York, 1991.