High-dimensional proportionality test of two covariance matrices and its application to gene expression data

With the development of modern science and technology, more and more high-dimensional data appear in the application fields. Since the high dimension can potentially increase the complexity of the covariance structure, comparing the covariance matrices among populations is strongly motivated in high-dimensional data analysis. In this article, we consider the proportionality test of two high-dimensional covariance matrices, where the data dimension is potentially much larger than the sample sizes, or even larger than the squares of the sample sizes. We devise a novel high-dimensional spatial rank test that has much-improved power than many existing popular tests, especially for the data generated from some heavy-tailed distributions. The asymptotic normality of the proposed test statistics is established under the family of elliptically symmetric distributions, which is a more general distribution family than the normal distribution family, including numerous commonly used heavy-tailed distributions. Extensive numerical experiments demonstrate the superiority of the proposed test in terms of both empirical size and power. Then, a real data analysis demonstrates the practicability of the proposed test for high-dimensional gene expression data.