Sankhya: The Indian Journal of Statistics
On the Distribution of the Product and Ratio of Independent Generalized Gamma-Ratio Random Variables
Carlos A. Coelho and Jo\~ao T. Mexia, The New University of Lisbon, Portugal
SUMMARY. Using a decomposition of the characteristic function of the logarithm of the product of independent generalized gamma-ratio random variables (r.v.'s), we obtain explicit expressions for both the probability density and cumulative distribution functions of the product of independent r.v.'s with generalized $F$ or generalized gamma-ratio distributions in the form of particular mixtures of generalized Pareto and inverted Pareto distributions. The expressions obtained do not involve any unsolved integrals and are convenient for computer implementation. By considering power parameters which are not required to be positive, we were able to obtain, as particular cases, not only the distributions for the product of folded T and folded Cauchy r.v.'s but also for the ratio of two independent products of generalized gamma-ratio r.v.'s. Theoretical applications of the results as well as simulations are presented.
AMS (2000) subject classification. Primary 62E15; secondary 62E10.
Key words and phrases. Particular mixtures, Pareto and inverted Pareto distributions, GIG distribution, sum of exponentials, difference of exponentials, folded T, folded Cauchy, beta prime, beta second kind.