A nonparametric test for elliptical distribution based on kernel embedding of probabilities

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 asymptotic distributions 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 also develop the concentration bounds of the test statistic, allowing the dimension to grow with the sample size, and further establish the consistency in this high-dimension setting. We compare our method with several existing methods via simulation studies, and apply our test to a SENIC dataset with and without a transformation aimed to achieve ellipticity.