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Zbl 1117.62017
Coelho, Carlos A.; Alberto, Rui P.; Grilo, Lu\'is M.
A mixture of generalized integer gamma distributions as the exact distribution of the product of an odd number of independent Beta random variables: applications. (English)
[J] J. Interdiscip. Math. 9, No. 2, 229-248 (2006). ISSN 0972-0502

Summary: 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 $\Lambda$ 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 $\Lambda$ 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 carried out.

MSC 2000:: *62E15 Exact distribution theory in statistics
62G10 Nonparametric hypothesis testing
62E10 Structure theory of statistical distributions

Keywords: independent gamma random variables; mixtures; mixture exponential distributions; mixture generalized integer gamma distributions; improper weights; generalized Wilks lambda; likelihood ratio statistic

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