Development and Comparative Study of two Near-exact Approximations to the Distribution of the Product of an Odd Number of Independent Beta Random Variables
by
Luís Miguel Grilo
Polytechnic Institute of Tomar, Portugal
Coauthors: Carlos Agra Coelho

Using the Generalized Near-Integer Gamma distribution, recently developed, as a basis, two different near-exact approximations to the distribution of the product of an odd number of particular independent Beta random variables is presented. By factoring the characteristic function of the logarithm of the product of an odd number of independent Beta random variables with parameters yielding particular relationships, adequate to be used as a basis for distributions of several statistics used in Multivariate Analysis, and then by replacing a suitably chosen factor of that characteristic function by an adequate asymptotic result it is possible to obtain what we call a near-exact characteristic function, which by inversion gives rise to the near-exact approximation to the exact distribution. The method used and results obtained are then applied in obtaining near-exact approximations to the distribution of the generalized Wilks Lambda statistic when there are two or more sets with an odd number of variables. Depending on the asymptotic result used to replace the chosen parts of the characteristic function, one may obtain different near-exact approximations. Moments from the two near-exact approximations developed are compared with the exact ones and the two approximations are compared with each other, namely in terms of moments and quantiles.

Date received: July 7, 2003