In this paper 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 Λ 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 Λ 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.

Summary Two different medical measurement methods, which usually do not produce exactly the same results, are used to analyse the serum levels of folic acid in a blood sample. We assess the (dis)agreement of the available data in order to replace the old method (the reference method, which involves a lot of human intervention) with the new one (which uses mostly machines), without causing problems in clinical interpretation. The 95% limits of agreement are estimated, before and after a logarithmic transformation, and an appropriate use of regression and a nonparametric approach are also considered. The application of these different statistical techniques is very useful and easily interpreted by medical researchers, but the results obtained do not provide confidence that the new method can be used in place of the old one for clinical purposes.

For the case where at least two sets have an odd number of variables we do not have the exact distribution of the generalized Wilks Lambda statistic in a manageable form, adequate for manipulation. In this article, we develop a family of very accurate near-exact distributions for this statistic for the case where two or three sets have an odd number of variables. We first express the exact characteristic function of the logarithm of the statistic in the form of the characteristic function of an infinite mixture of Generalized Integer Gamma distributions. Then, based on truncations of this exact characteristic function, we obtain a family of near-exact distributions, which, by construction, match the first two exact moments. These near-exact distributions display an asymptotic behaviour for increasing number of variables involved. The corresponding cumulative distribution functions are obtained in a concise and manageable form, relatively easy to implement computationally, allowing for the computation of virtually exact quantiles. We undertake a comparative study for small sample sizes, using two proximity measures based on the Berry-Esseen bounds, to assess the performance of the near-exact distributions for different numbers of sets of variables and different numbers of variables in each set.