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

Pathway idea is a switching mechanism by which one can go from one functional form to another, and to yet another. It is shown that through a parameter α, called the pathway parameter, one can connect generalized type-1 beta family of densities, generalized type-2 beta family of densities, and generalized gamma family of densities, in the scalar as well as the matrix cases, also in the real and complex domains. It is shown that when the model is applied to physical situations then the current hot topics of Tsallis statistics and superstatistics in statistical mechanics become special cases of the pathway model, and the model is capable of capturing many stable situations as well as the unstable or chaotic neighborhoods of the stable situations and transitional stages. The pathway model is shown to be connected to generalized information measures or entropies, power law, likelihood ratio criterion or λ−criterion in multivariate statistical analysis, generalized Dirichlet densities, fractional calculus, Mittag-Leffler stochastic process, Krätzel integral in applied analysis, and many other topics in different disciplines. The pathway model enables one to extend the current results on quadratic and bilinear forms, when the samples come from Gaussian populations, to wider classes of populations.Axioms 2015, 4 531

Here an extended form of the reaction rate probability integral, in the case of nonresonant thermonuclear reactions with the depleted tail and the right tail cut off, is considered. The reaction rate integral then can be looked upon as the inverse of the convolution of the Mellin transforms of Tsallis type statistics of nonextensive statistical mechanics and stretched exponential as well as that of superstatistics and stretched exponentials. The differential equations satisfied by the extended probability integrals are derived. The idea used is a novel one of evaluating the extended integrals in terms of some special functions and then by invoking the differential equations satisfied by these special functions. Some special cases of limiting situations are also discussed.

Abstract:In this paper, I consider multivariate analogues of the extended gamma density, which will provide multivariate extensions to Tsallis statistics and superstatistics. By making use of the pathway parameter β, multivariate generalized gamma density can be obtained from the model considered here. Some of its special cases and limiting cases are also mentioned. Conditional density, best predictor function, regression theory, etc., connected with this model are also introduced.