Permutation testing for goodness-of-fit and stochastic ordering with multivariate mixed variables

Permutation tests are highly versatile non-parametric procedures that can be used to address a wide set of statistical problems, without strict assumptions on data distribution. The Non-Parametric Combination (NPC) procedure has been proposed in the multivariate context to combine the results of several univariate permutation tests. This work demonstrates the flexibility and power of the procedure with a focus on Goodness-of-Fit and the comparison of C>2 different distributions, and includes the particular case of stochastic ordering problems. For each problem, we propose a different extension of the NPC procedure and suitable solutions for contexts in which the data are not solely continuous or ordinal, but also mixed. Additionally, the paper shows how these procedures can work with small total sample size n, even when n is lower than the number of variables V, and how a higher value of V has a positive effect on the power of the tests.