Distribution of a Product and the Structural Set Up of Densities

Abstract

This paper is a study of statistical distributions by analyzing the structure of density functions. This work was motivated by the paper [10] in which the distribution of the product of two non-central chi-square variables is obtained. Their problem arose from a problem in the physical sciences connected with the theory of spinstabilized rockets. It is pointed out in [10] that the original problem was to obtain the distribution of the product of a central Raleigh and a non-central Raleigh variable. By examining the structure of the density function of non-central Raleigh or non-central chi-square it is apparent that the density is the product of the special cases of two special functions. In [8, chapter 2] a lengthy treatment of Raleigh and associated distributions is given. We do not know any other particular problem in the physical sciences, but from the structural property of the problem pointed out in [10] it is quite likely that there may be a number of such problems in the physical sciences or in other disciplines, which are to be tackled. Hence we will give the distribution of the product of two independent stochastic variables whose density functions can be expressed as the product of any two special functions. Products of $H$-functions are used so that almost all classical density functions (central or non-central) will be taken care of, since $H$-functions are the most generalized special functions. Since a number of types of factors can be absorbed inside the $H$-function we think that all the distributions which are frequently used in the statistical theory of distributions will be included in this problem that we discuss here, with some modifications in some cases. Several special cases are pointed out so that one can easily get the distribution of the product or ratio of independent stochastic variables whose density functions are products of special functions. The result in [10] is obtained as a special case and further, the distributions of the products of two non-central $F$ simple and multiple correlation coefficients are pointed out for the sake of mathematical interest because, structurally, the density functions in these cases belong to different categories. Since several properties of $H$-functions are available in the literature, it is easy to study other properties or to compute percentage points of the product distribution discussed in this paper. The approach of examining the structure of densities may simplify the problem of obtaining distributions of several statistics.