Erdelyi-Kober Fractional Integral Operators from a Statistical Perspective -I

TL;DR

This paper explores the statistical interpretation of Erdelyi-Kober fractional integral operators, linking them to densities of products and ratios of independent positive random variables, and generalizes these operators through various mathematical frameworks.

Contribution

It provides a novel statistical perspective on Erdelyi-Kober operators, connecting them to Mellin convolutions and introducing a broad class of generalized operators with statistical interpretations.

Findings

Kober operators correspond to densities of products and ratios of independent variables.

Generalizations include pathway and hypergeometric series-based operators.

Standard operators are special cases within the generalized framework.

Abstract

In this article we examine the densities of a product and a ratio of two real positive scalar random variables $x_1$ and $x_2$, which are statistically independently distributed, and we consider the density of the product $u_1=x_1x_2$ as well as the density of the ratio $u_2={{x_2}\over{x_1}}$ and show that Kober operator of the second kind is available as the density of $u_1$ and Kober operator of the first kind is available as the density of $u_2$ when $x_1$ has a type-1 beta density and $x_2$ has an arbitrary density. We also give interpretations of Kober operators of the second and first kind as Mellin convolution for a product and ratio respectively. Then we look at various types of generalizations of the idea thereby obtaining a large collection of operators which can all be called generalized Kober operators. One of the generalizations considered is the pathway idea where one can move from one family of operators to another family and yet another family and eventually end up with an exponential form. Common generalizations in terms of a Gauss' hypergeometric series is also given a statistical interpretation and put on a more general structure so that the standard generalizations given by various authors, including Saigo operators, are given statistical interpretations and are derivable as special cases of the general structure considered in this article.