On Finding Geodesic Equation of Two Parameters Logistic Distribution

William W. S. Chen

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

DOI: 10.4236/am.2015.612189                

Abstract

In this paper, we used two different algorithms to solve some partial differential equations, where these equations originated from the well-known two parameters of logistic distributions. The first method was the classical one that involved solving a triply of partial differential equations. The second approach was the well-known Darboux Theory. We found that the geodesic equations are a pair of isotropic curves or minimal curves. As expected, the two methods reached the same result.

Keywords

Darboux Theory, Differential Geometry, Geodesic Equation, Isotropic Curves, Logistic Distribution, Minimal Curves, Partial Differential Equation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Mitchell, A.F.S. (1992) Estimative and Predictive Distances. Test, 1, 105-121. http://dx.doi.org/10.1007/BF02562666
[2]	Mitchell, A.F.S. and Krzanowski, W.J. (1985) The Mahalanobis Distance and Elliptic Distributions. Biometrika, 72, 464-467. http://dx.doi.org/10.1093/biomet/72.2.464
[3]	Kass, R.E. and Vos, P.W. (1997) Geometrical Foundations of Asymptotic Inference. John Wiley & Sons, Inc. http://dx.doi.org/10.1002/9781118165980
[4]	Amari, S.-I. (1990) Differential-Geometrical Methods in Statistics. Springer, New York.
[5]	Jensen, U. (1995) Review of “The Derivation and Calculation of Rao Distances with an Application to Portfolio Theory”. In: Maddala, P., Phillips, G.S. and Srinivasan, T., Eds., Advances in Econometrics and Quantitative Economics: Essays in Honor of C.R. Rao, Blackwell, Cambridge, 413-462.
[6]	Struik, D.J. (1961) Lectures on Classical Differential Geometry. 2nd Edition, Dover Publications, Inc.
[7]	Grey, A. (1993) Modern Differential Geometry of Curves and Surfaces. CRC Press, Inc., Boca Raton.
[8]	Chen, W.W.S. (2014) A Note on Finding Geodesic Equation of Two Parameters Gamma Distribution. Applied Mathematics, 5, 3511-3517. www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2014.521328
[9]	Chen, W.W.S. (2014) A Note on Finding Geodesic Equation of Two Parameter Weibull Distribution. Theoretical Mathematics & Applications, 4, 43-52.
[10]	Balakrishnan, N. and Nevzorov, V.B. (2003) A Primer on Statistical Distributions. John Wiley & Sons, Inc.
[11]	Gradshteyn, I.S., Ryzhik, I.M. and Jeffrey, A. (1994) Table of Integrals, Series, and Products. 5th Edition, Academic Press.