A mixture of Generalized Integer Gamma distributions as the exact distribution of the product of an odd number of independent Beta random variables: applications

Abstract

In this paper we show first how the distribution of the logarithm of a random variable with a Beta distribution may be expressed either as a mixture of Gamma distributions or as a mixture of Generalized Integer Gamma (GIG) distributions and then how the exact distribution of the product of an odd number of independent Beta random variables whose first parameter evolves by 1/2 and whose second parameter is the half of an odd integer may be expressed as a mixture of GIG distributions. Some particularities of these mixtures are analysed. The results are then used to obtain the exact distribution of the logarithm of the Wilks Λ statistic to test the independence of two sets of variables, both with an odd number of variables, and the exact distribution of the logarithm of the generalized Wilks Λ statistic to test the independence of several sets of variables, in the case where two or three of them have an odd number of variables. A discussion of relative advantages and disadvantages of the use of the exact versus near-exact distributions is carried out.