Multivariate Tri-Aspect Non-Parametric Testing

Permutation tests are prized for their lack of assumptions concerning distribution of underlying populations. The (usual) permutation test for the two-sample location problem based on comparison of sample means is generally effective with regular, roughly symmetric, unimodal, and light-tailed distributions, whereas it might not be so with highly asymmetric and/or heavy-tailed distributions. Another drawback is that it is not consistent for distributions for which first and second moments do not exist. Marozzi [Marozzi, M., 2004, A bi-aspect nonparametric test for the two-sample location problem. Computational Statistics and Data Analysis, 44, 639–648.] proposed a bi-aspect non-parametric test for comparing two populations obtained by non-parametric combining the usual permutation test (which addresses the numerical aspect Xi ) and a test based on comparison of frequencies over the pooled median (which addresses the categorical aspect related to the comparison of sample units with the pooled sample median). Unlike the usual permutation test, the bi-aspect test is consistent for every distribution and is very powerful with highly-skewed and/or heavy-tailed distributions. In the paper, the bi-aspect testing idea is extended by also considering the aspect based on ranks, with the role of third aspect. A simulation study with many sample size and distribution settings shows that the triaspect test is more powerful than the bi-aspect one. Moreover, the multivariate problem is addressed and formal proofs of exactness, unbiasedness, and consistency are given.