Near-exact distributions for the generalized Wilks Lambda criterion based on truncations of the exact characteristic function
by
Luis Miguel Grilo
Department of Mathematics of the Polytechnic Institute of Tomar, Center for Mathematics and its Applications to the Faculty of Sciences and Technology, New University of Lisbon, Portugal
Coauthors: Carlos Agra Coelho - Department of Mathematics and Center for Mathematics and its Applications Faculty of Sciences and Technology, New University of Lisbon, Portugal

Near-exact distributions for the product of an odd number of independent Beta random variables are developed, based on truncations of the exact characteristic function, which involves an infinite mixture of Generalized Integer Gamma distributions. By direct application of these results and once again based on truncations of the exact characteristic function, we obtain near-exact distributions for the generalized Wilks Lambda statistic used in testing the independence of several groups of variables, when a maximum of three of these groups have an odd number of variables (being the statistic used to test the independence of two groups of variables, both with an odd number of variables, a particular case). These near-exact distributions are finite mixtures of Generalized Integer Gamma distributions and Generalized Near-Integer Gamma distributions, and are relatively easy to implement computationally, allowing for the computation of near-exact quantiles which may indeed be regarded as virtually exact, given the good convergence properties of the series involved, mainly when the difference between sample size and the overall number of variables involved is rather small.

Keywords and phrases: Product of Beta random variables, sum of Gamma random variables, mixtures, generalized Wilks Lambda statistic, proximity measures.

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Date received: July 12, 2007