Distribution-Free Signs and Ranks via Optimal Transport under Multivariate Symmetry and Application to One-Sample Location Testing
利用最优传输理论，提出一种统一的框架，在多元对称性下构造无分布广义符号、绝对秩和符号秩，并发展出多元符号检验和Wilcoxon符号秩检验，在有限样本中精确无分布且渐近正态，对位置偏移尤其有效。
We propose a novel and unified framework for distribution-free testing under multivariate symmetry (that includes central symmetry, sign symmetry, spherical symmetry, etc.) based on the theory of optimal transport. Our approach leads to notions of distribution-free generalized multivariate signs, absolute ranks and signed-ranks. As a consequence, we develop analogues of the sign and Wilcoxon signed-rank tests that share many of the appealing properties of their one-dimensional counterparts. In particular, the proposed tests are exactly distribution-free in finite samples with an asymptotic normal limit, and adapt to various notions of multivariate symmetry. We study the consistency of the proposed tests and their behavior under local alternatives, and show that the proposed generalized Wilcoxon signed-rank (GWSR) test is particularly powerful against location shift alternatives. We show that in a large class of such models, our GWSR test suffers from no loss in (asymptotic) efficiency, when compared to Hotelling’s T2 test, despite being nonparametric and exactly distribution-free. An appropriately score transformed version of the GWSR statistic leads to a locally asymptotically optimal test. Further, our method can be readily used to construct distribution-free confidence sets for the center of symmetry. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.