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 article proposes a new approach that uses 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 dataset is provided as well. Supplementary materials for this article are available online.

置换检验（Permutation tests）适用于检验统计假设的场景，此时检验统计量的分布复杂或无法获取。在部分场景中，所研究的检验统计量为多元形式，多重检验问题便是其中一类典型示例。针对此类场景，现有多元置换检验通常通过所谓的组合函数，对偏置换p值向量进行合适的一维转换以构建检验方法。本文提出了一种全新方法，该方法引入了离散最优测度传输（discrete optimal measure transportation）概念。最终得到的单一p值，由置换后的多元检验统计量的经验中心向外分布函数计算得出。该方法无需计算部分p值，且易于实现。此外，该方法支持计算并解释多元检验统计量的各分量对整体非一致性评分与原假设拒绝决策的贡献。除上述方法外，本文还将测度传输应用于偏p值向量，作为经典组合函数的替代方案。在蒙特卡洛（Monte Carlo）研究中，本文通过多个实际示例将这两种技术与标准方法进行了对比。此外，本文还提供了该方法在函数型数据集（functional dataset）上的应用案例。本文的补充材料可在线获取。