  Permutation tests enable testing statistical hypotheses in situations when
the distribution of the test statistic is complicated or not available. In some
situations, the test statistic under investigation is multivariate, with the
multiple testing problem being an important example. The corresponding
multivariate permutation tests are then typically based on a
suitableone-dimensional transformation of the vector of partial permutation
p-values via so called combining functions. This paper proposes a new approach
that utilizes the optimal measure transportation concept. The final single
p-value is computed from the empirical center-outward distribution function of
the permuted multivariate test statistics. This method avoids computation of
the partial p-values and it is easy to be implemented. In addition, it allows
to compute and interpret contributions of the components of the multivariate
test statistic to the non-conformity score and to the rejection of the null
hypothesis. Apart from this method, the measure transportation is applied also
to the vector of partial p-values as an alternative to the classical combining
functions. Both techniques are compared with the standard approaches using
various practical examples in a Monte Carlo study. An application on a
functional data set is provided as well.
