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

While the problem of testing multivariate normality has received considerable attention in the classical low-dimensional setting where the sample size $n$ 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 $d$ is of comparable or larger order than $n$. This paper studies the hypothesis testing problem of determining 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 under a broad regime. To this end, we propose a new class of computationally efficient tests which can be regarded as a high-dimensional adaptation of the classical radial approach to testing normality. A key member of this class is a range-type test which, under a very general rate of growth of $d$ with respect to $n$, is proven to achieve both 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 our test compared to existing methods, and two gene expression applications demonstrate the effectiveness of our procedure for detecting violations of multivariate normality which are of potentially practical significance.