High-Dimensional Invariant Tests of Multivariate Normality Based on Radial Concentration

Abstract: While the problem of testing multivariate normality has received a considerable amount of attention in the classical low-dimensional setting where the number of samples $n$ is much larger than the feature dimension $d$ of the data, there is presently a dearth of existing tests which are valid in the high-dimensional setting where $d$ may be of comparable or larger order than $n$. This paper studies the hypothesis-testing problem regarding whether $n$ i.i.d. samples are generated from a $d$-dimensional multivariate normal distribution in settings where $d$ grows with $n$ at some rate. To this end, we propose a new class of tests which can be regarded as a high-dimensional adaptation of the classical radial-based approach to testing multivariate normality. A key member of this class is a range-type test statistic which, under a very general rate of growth of $d$ with respect to $n$, is proven to achieve both valid type I error-control and consistency for three important classes of alternatives; namely, finite mixture model, non-Gaussian elliptical, and leptokurtic alternatives. Extensive simulation studies demonstrate the superiority of the proposed testing procedure compared to existing methods, and two gene expression applications are used to demonstrate the effectiveness of our methodology for detecting violations of multivariate normality which are of potentially critical practical significance.