A Note on Finding Geodesic Equation of Two Parameters Gamma Distribution

William W S Chen

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].

Highlights

Rao [3] introduced a Riemannian metric over the space of a parametric family of probability distribution
He proposed the minimized distance induced by the metric as a measure of dissimilarity between probability distribution
Mitchell [4] worked on statistical manifolds of univariate or multivariate elliptic distributions and found the α Gaussian Curvature and geodesics for the univariate elliptic class

Summary

Introduction

Rao [3] introduced a Riemannian metric over the space of a parametric family of probability distribution He proposed the minimized distance induced by the metric as a measure of dissimilarity between probability distribution. Chen and Kotz [7] have studied the Riemannian structure of the three-parameter gamma distribution. (2014) A Note on Finding Geodesic Equation of Two Parameters Gamma Distribution. S. Chen ties in classical differential geometry to find the geodesic equation of gamma distribution. Chen ties in classical differential geometry to find the geodesic equation of gamma distribution Applying these results with the Darboux Theory [8] helps us find a natural solution of the Geodesic Equation. We will apply the traditional technique of finding the Geodesic Equation of the Gamma Manifold in order to compare it with the Darboux Approach.

The Geodesic Equation

List the Fundamental Tensor