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