Product of independent generalised gamma random variables

Abstract

The generalised gamma distribution has received much attention due to its exibility and also for having some well-known distributions as special cases. This study originates from a statistic de ned as the ratio of products of independent generalised gamma random variables and shows that it can be represented as the product of independent generalised gamma random variables with some re-parametrisation. By decomposing the character- istic function of the negative logarithm of the statistic and then using the distribution of the di¤erence of two independent generalized integer gamma random variables as a basis, accurate and computationally appealing near-exact distributions are derived for the statis- tic. In the process, a new exible 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 be- tween cumulative distribution functions; also, by comparison with the exact distribution, empirical distribution and with an approximation developed using a di¤erent method and which can only be applied in some particular cases.