HOW TO DEVELOP NEAR-EXACT DISTRIBUTIONS: THE CASE OF THE DISTRIBUTION OF THE GENERALIZED WILKS LAMBDA STATISTIC

Abstract: In developing near-exact distributions we try to keep the most part of the exact characteristic function of the random variable of interest unchanged and replace the smaller remaining part by an asymptotic result in such a way that the resulting characteristic function is easier to invert than the unmodified one. This way, it’s possible to obtain what we call a near-exact distributions based on factorization of the exact characteristic function or to obtain near-exact distributions based on truncations of the exact characteristic function. These near-exact distributions are usually obtained under the form of Generalized Integer Gamma (GIG) distributions and Generalized Near‑Integer Gamma (GNIG) distributions or mixtures of these distributions and they are relatively easy to implement computationally, allowing for the computation of near‑exact quantiles. They are obtained in a manageable form and by construction some of the first moments are equal to the exact ones.

We start to express the generalized Wilks Lambda statistic, used to test the independence of several sets of variables, when a maximum of three of these sets have an odd number of variables (being the statistic used to test the independence of two sets of variables, both with an odd number of variables, a particular case), as a infinite mixture of GIG distributions. Then we develop a family of near-exact distributions for the generalized Wilks Lambda statistic, based on truncations of the exact characteristic function, and they are finite mixtures of GIG distributions and GNIG distributions. We assess the proximity between these near-exact distributions and the exact distribution, for different number of sets, different number of variables in each set and different sample sizes, by using two measures based on the Berry‑Esseen bounds. We also compared them with asymptotic distributions and near-exact distributions based on factorizations of the exact characteristic function. The family members of the near‑exact distribution, based on truncations, equate the two first exact moments andallow us to obtain near-exact quantiles, which may indeed be regarded as virtually exact, given the good convergence properties of the series involved. These near-exact distributions show an excellent performance and they are particularly adequate and useful for cases where the difference between sample size and the overall number of variables involved is small.

Keywords and phrases: Beta and Gamma random variables, Sum of Gamma random variables, mixtures, proximity measures.