The Exact and Near-Exact Distributions for the Wilks Lambda Statistic Used in the Test of Independence of Two Sets of Variables

Abstract

Synoptic We develop the exact distribution of the Wilks Lambda statistic to test the independence of two sets of variables, both with an odd number of variables, under the form of an infinite mixture of Generalized Integer Gamma distributions. Based on truncations of the exact characteristic function, for the product of independent Beta random variables, we obtain near-exact distributions for such product and then by direct application of these results, and once again based on truncations, we develop near-exact distributions for the Wilks Lambda statistic. These near-exact distributions are finite mixtures of Generalized Integer Gamma and Generalized Near-Integer Gamma distributions. By construction, the two first moments of these approximations are equal to the exact moments. These distributions are manageable and 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. We assess the proximity between these near-exact di stributions and the exact distribution by using two measures based on the Berry-Esseen bounds.