Testing the equality of two high-dimensional spatial sign covariance matrices

This paper is concerned with testing the equality of two high-dimensional spatial sign covariance matrices with applications to testing the proportionality of two high-dimensional covariance matrices. It is interesting that these two testing problems are completely equivalent for the class of elliptically symmetric distributions. This paper develops a new test for testing the equality of two high-dimensional spatial sign covariance matrices based on the Frobenius norm of the difference between two spatial sign covariance matrices. The asymptotic normality of the proposed testing statistic is derived under the null and alternative hypotheses when the dimension and sample sizes both tend to infinity. Moreover, the asymptotic power function is also presented. Simulation studies show that the proposed test performs very well in a wide range of settings and can be allowed for the case of large dimensions and small sample sizes.