Two-Sample Covariance Inference in High-Dimensional Elliptical Models

Abstract: We propose a two-sample test for large-dimensional covariance matrices in generalized elliptical models. The test statistic is based on a U-statistic estimator of the squared Frobenius norm of the difference between the two population covariance matrices. This statistic was originally introduced by Li and Chen (2012, AoS) for the independent component model. As a key theoretical contribution, we establish a new central limit theorem for the U-statistics under elliptical data, valid under both the null and alternative hypotheses. This result enables asymptotic control of the test level and facilitates a power analysis. To the best of our knowledge, the proposed test is the first such method to be supported by theoretical guarantees for elliptical data. Our approach imposes only mild assumptions on the covariance matrices and does neither require sparsity nor explicit growth conditions on the dimension-to-sample-size ratio. We illustrate our theoretical findings through applications to both synthetic and real-world data.