  Many statistical methodologies for high-dimensional data assume the
population is normal. Although a few multivariate normality tests have been
proposed, to the best of our knowledge, none of them can properly control the
type I error when the dimension is larger than the number of observations. In
this work, we propose a novel nonparametric test that utilizes the nearest
neighbor information. The proposed method guarantees the asymptotic type I
error control under the high-dimensional setting. Simulation studies verify the
empirical size performance of the proposed test when the dimension grows with
the sample size and at the same time exhibit a superior power performance of
the new test compared with alternative methods. We also illustrate our approach
through two popularly used data sets in high-dimensional classification and
clustering literatures where deviation from the normality assumption may lead
to invalid conclusions.
