Gamma distribution and extensions by using pathway idea

Abstract

Pathway model, Generalized gamma, Extended gamma, Inverse Gaussian, H-function,

An Overview of the Pathway Idea and Its Applications in Statistical and Physical Sciences

“…From the path created by α we note that we obtain densities with thicker or thinner tail compared to generalized gamma density. From Figure 1 (Right) we can see that when α moves from −∞ to 1, the curve becomes thicker tailed and less peaked, see ( [8,9]). Note that α is the most important parameter here for enabling one to more than one family of functions.…”

“…Since the density is unique, in whatever way we evaluate the density we should get the same function or the functions will be identical. We can evaluate the density of u by the method of inverse Mellin transform, see [9]. Comparing the density obtained in these two different methods, we will get the Laplace transform of the pathway model given in Equation (2) as an H-function…”

Multivariate Extended Gamma Distribution

“…Here we consider the multivariate case of the extended gamma density of the form (2). For X i ≥ 0, i = 1, 2, .…”

“…From the path created by β we note that we obtain densities with thicker or thinner tail compared to generalized gamma density. Observe that for β < 1, writing β − 1 = −(1 − β) in Equation (2) produce generalized type-1 beta form, which is given by…”

“…, is the normalizing constant (see [2]). From the above graph, one can see the movement of the extended gamma density denoted by g 1 (x) towards the generalized gamma density, for various values of the pathway parameter β. Beck and Cohen's superstatistics belong to the case (2) [3,4].…”