The sign test (Arbuthnott, 1710) and the Wilcoxon signed-rank test (Wilcoxon, 1945) are among the first examples of a nonparametric test. These procedures -- based on signs, (absolute) ranks and signed-ranks -- yield distribution-free tests for symmetry in one-dimension. In this paper we propose a novel and unified framework for distribution-free testing of 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, 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 $T^2$ 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.

翻译：符号检验（Arbuthnott, 1710）和威尔科克森符号秩检验（Wilcoxon, 1945）是非参数检验的早期范例。这些基于符号、（绝对）秩和符号秩的程序，为一维对称性提供了无分布检验。本文基于最优运输理论，提出了一种新颖且统一的多元对称性（包括中心对称性、符号对称性、球面对称性等）无分布检验框架。我们的方法引出了无分布广义多元符号、秩和符号秩的概念。由此，我们开发了符号检验和威尔科克森符号秩检验的类似物，它们共享一维对应检验的许多优良性质。具体而言，所提出的检验在有限样本中精确无分布且具有渐近正态极限，并能适应多种多元对称性概念。我们研究了所提检验的一致性及其在局部备择假设下的行为，并表明所提出的广义威尔科克森符号秩检验在位置偏移备择假设下特别有效。我们证明了，在一大类此类模型中，尽管我们的GWSR检验是非参数且精确无分布的，但与霍特林$T^2$检验相比，其在（渐近）效率上没有任何损失。对GWSR统计量进行适当得分变换后，可得到局部渐近最优检验。此外，我们的方法可直接用于构建对称中心的无分布置信集。