Multivariate quantile-based permutation tests with application to functional data

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 suitable one-dimensional transformation of the vector of partial permutation p-values via so called combining functions. This paper proposes a new approach that utilizes the discrete 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 overall 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 to the standard approaches using various practical examples in a Monte Carlo study. An application to a functional data set is provided as well.

置换检验（Permutation tests）可在检验统计量分布复杂或未知的场景下，用于统计假设检验。在部分场景中，待研究的检验统计量为多元形式，多重检验问题便是典型示例之一。相应的多元置换检验通常基于通过所谓组合函数对部分置换p值向量进行的恰当一维变换。本文提出一种全新方法，该方法利用离散最优测度传输（discrete optimal measure transportation）概念。最终的单一p值可通过置换后多元检验统计量的经验中心向外分布函数计算得到。该方法无需计算部分p值，且易于实现。此外，该方法可计算并解释多元检验统计量各分量对总体不一致性得分以及原假设拒绝决策的贡献程度。除上述方法外，本文还将测度传输应用于部分p值向量，以此作为经典组合函数的替代方案。本文通过蒙特卡洛模拟研究中的多个实际示例，将这两种技术与标准方法进行对比；此外还提供了该方法在函数型数据集上的应用案例。