Products of ratios of gamma functions - An application to the distribution of the test statistic for testing the equality of covariance matrices

An expansion of the ratio of two gamma functions is used to obtain an adequate representation for the product of ratios of gamma functions which enables the derivation of approximating expansions for the distribution of the test statistic used for testing the equality of several covariance matrices. These asymptotic approximations are based on mixtures of gamma distributions and are obtained using a new expression of the Fourier transform of the density function of the logarithm of the test statistic and also on a matching moments technique. Numerical studies show that these approximations are simple, precise and easy to implement. A computational module, developed for Mathematica software, is presented together with an example of application.