Testing Elliptical Models in High Dimensions

Due to the broad applications of elliptical models, there is a long line of research on goodness-of-fit tests for empirically validating them. However, the existing literature on this topic is generally confined to low-dimensional settings, and to the best of our knowledge, there are no established goodness-of-fit tests for elliptical models that are supported by theoretical guarantees in high dimensions. In this article, we propose a new goodness-of-fit test for this problem, and our main result shows that the test is asymptotically valid when the dimension and sample size diverge proportionally. Remarkably, it also turns out that the asymptotic validity of the test requires no assumptions on the population covariance matrix. With regard to numerical performance, we confirm that the empirical level of the test is close to the nominal level across a range of conditions, and that the test is able to reliably detect non-elliptical distributions. Moreover, when the proposed test is specialized to the problem of testing normality in high dimensions, we show that it compares favorably with a state-of-the-art method, and hence, this way of using the proposed test is of independent interest. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.