Erdelyi-Kober fractional integral operators from a statistical perspective (I)A. M. Mathai, H. J. Haubold
In this article we examine the densities of a product and a ratio of two real positive scalar random variables x1 and x2,
which are statistically independently distributed, and we consider the density of the product u1=x1x2 as well as the density of
the ratio u2=x2/x1 and show that Kober operator of the second kind is available as the density of
u1 and Kober operator of the first kind is available as the density of u2 when x1 has a type-1 beta density and x2 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.
Tbilisi Mathematical Journal, Vol. 10(1) (2017), pp. 145-159 |