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Application of the exact statistical inference frequently leads to anon-standard probability distributions of the considered estimators or teststatistics. The exact distributions of many estimators and test statistics canbe specified by their characteristic functions. Typically, distribution of manyestimators and test statistics can be structurally expressed as a linearcombination or product of independent random variables with known distributionsand characteristic functions, as is the case for many standard multivariatetest criteria. The characteristic function represents complete characterizationof the distribution of the random variable. However, analytical inversion ofthe characteristic function, if possible, frequently leads to a complicated andcomputationally rather strange expressions for the corresponding distributionfunction (CDF/PDF) and the required quantiles. As an efficient alternative,here we advocate to use the well-known method based on numerical inversion ofthe characteristic functions --- a method which is, however, ignored in popularstatistical software packages. The applicability of the approach is illustratedby computing the exact distribution of the Bartlett's test statistic fortesting homogeneity of variances in several normal populations and the Wilks's$\Lambda$-distribution used in multivariate hypothesis testing.

Exact distribution of selected multivariate test criteria by numerical inversion of their characteristic functions