The ratio of independent generalized gamma random variables with applications

Abstract

This paper originates from the interest in the distribution of a statistic defined as the ratio of independent generalized gamma random variables. It is shown that it can be represented as the product of independent generalized gamma random variables with some reparametrization. By decomposing the characteristic function of the negative natural logarithm of the statistic and by using the distribution of the difference of two independent generalized integer gamma random variables as a basis, accurate and computationally appealing near-exact distributions are derived for this statistic. In the process, a new parameter is introduced in the near-exact distributions, which allows to control the degree of precision of these approximations. Furthermore, the performance of the near-exact distributions is assessed using a measure of proximity between cumulative distribution functions and by comparison with the exact and empirical distributions. We illustrate the use of the proposed approximations on the distribution of the ratio of generalized variances in a multivariate multiple regression setting and with an example of application related with single-input single-output networks. The proposed results ensure less computing time and stability in results as well.