A mixture of Generalized Integer Gamma distributions as the exact distribution of the product of an odd number of independent Beta random variables with first parameter evolving by 1/2. Applications.
by
Carlos A. Coelho
Dep. of Mathematics, Institute of Agriculture Technology, Lisbon, Portugal
Coauthors: Rui P. Alberto (Bela Vista High School, Setubal, Portugal), Luis M. Grilo (Polytechnic Institute of Tomar)

First we show 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 Wilks Lambda statistic to test the independence of two sets of variables, both with an odd number of variables, and the distribution of the generalized Wilks Lambda 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 then carried out.

Date received: September 10, 2003