Near-exact distributions, based on truncations of the exact distribution, for the Wilks Lambda statistic used in test of the independence of several groups of variables
by
Luis M. Grilo & Carlos A. Coelho
Polytechnic Institute of Tomar, Tomar, Portugal & The New University of Lisbon, Portugal

Based on the expression of the Logbeta distribution as an infinite mixture of Exponential distributions we were able, by considering a certain number of terms in the series associated with the characteristic function and making an adequate approximation for the rest, to develop near-exact distributions, for the exact distribution of the product of an odd number of independent Beta random variables, which is expressed as a mixture of Generalized Near Integer Gamma (GNIG) distributions. By direct application of these results and once again based on truncations, we obtain near-exact distributions for the exact distribution of the 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. These near exact distributions are finite mixtures of Generalized Integer Gamma (GIG) distributions and GNIG 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 the sample size and the overall number of variables involved is rather small. By construction, the two first moments of these near exact distributions are equal to the exact moments.

Keywords: Near-exact distributions, Beta random variables, Gamma random variables, mixtures, generalized Wilks Lambda statistic, measures of proximity.

Date received: July 20, 2006