Elliptical distribution is a basic assumption underlying many multivariate statistical methods. For example, in sufficient dimension reduction and statistical graphical models, this assumption is routinely imposed to simplify the data dependence structure. Before applying such methods, we need to decide whether the data are elliptically distributed. Currently existing tests either focus exclusively on spherical distributions, or rely on bootstrap to determine the null distribution, or require specific forms of the alternative distribution. In this paper, we introduce a general nonparametric test for elliptical distribution based on kernel embedding of the probability measure that embodies the two properties that characterize an elliptical distribution: namely, after centering and rescaling, (1) the direction and length of the random vector are independent, and (2) the directional vector is uniformly distributed on the unit sphere. We derive the null asymptotic distribution of the test statistic via von-Mises expansion, develop the sample-level procedure to determine the rejection region, and establish the consistency and validity of the proposed test. We apply our test to a SENIC dataset with and without a transformation aimed to achieve ellipticity.

翻译：椭圆分布是许多多元统计方法的基本假设。例如，在充分降维和统计图模型中，通常施加这一假设以简化数据依赖结构。在应用此类方法之前，我们需要判断数据是否服从椭圆分布。现有检验方法或仅关注球面分布，或依赖自助法确定零分布，或要求备择分布具有特定形式。本文提出一种基于概率测度核嵌入的通用非参数检验方法，该方法体现了椭圆分布的两个特征性质：即经中心化和重新缩放后，（1）随机向量的方向与长度相互独立，且（2）方向向量在单位球面上均匀分布。我们通过冯·米塞斯展开推导了检验统计量的零渐近分布，开发了确定拒绝域的样本层面程序，并证明了所提检验的一致性和有效性。我们将该检验应用于SENIC数据集，分别考察了未进行变换和经过旨在实现椭圆性变换后的结果。