  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.
