We develop some graph-based tests for spherical symmetry of a multivariate distribution using a method based on data augmentation. These tests are constructed using a new notion of signs and ranks that are computed along a path obtained by optimizing an objective function based on pairwise dissimilarities among the observations in the augmented data set. The resulting tests based on these signs and ranks have the exact distribution-free property, and irrespective of the dimension of the data, the null distributions of the test statistics remain the same. These tests can be conveniently used for high-dimensional data, even when the dimension is much larger than the sample size. Under appropriate regularity conditions, we prove the consistency of these tests in high dimensional asymptotic regime, where the dimension grows to infinity while the sample size may or may not grow with the dimension. We also propose a generalization of our methods to take care of the situations, where the center of symmetry is not specified by the null hypothesis. Several simulated data sets and a real data set are analyzed to demonstrate the utility of the proposed tests.

翻译：我们基于数据增广方法，开发了若干用于多元分布球对称性的图基检验。这些检验通过一种新的符号与秩概念构建，该概念沿一条优化路径计算，该路径基于增广数据集中观测值间两两相异性构建的目标函数优化而得。基于这些符号与秩的检验具有精确无分布特性，且无论数据维度如何，检验统计量的零分布始终保持不变。这些检验可便捷地应用于高维数据，即使维度远大于样本量亦适用。在适当的正则性条件下，我们证明了这些检验在高维渐近体系中的相合性，其中维度趋于无穷而样本量可与维度同步增长亦可独立固定。我们还提出了本方法的推广形式，以处理对称中心未由原假设指定的情况。通过多个模拟数据集与一个真实数据集的分析，验证了所提检验的实用性。