An integrated test on the linear structure of high-dimensional covariance matrices

In the analysis of complex data from various fields like finance, imaging processing, and biomedical applications, the assumption regarding the structure of covariance holds a crucial role in ensuring accurate and efficient statistical inferences. We study the problem of evaluating whether a high-dimensional covariance matrix conforms to a linear structure defined by the linear combination of a predefined set of matrices. We introduce an innovative testing approach by integrating two Frobenius-type statistics, which capture both the difference and ratio between the unknown covariance matrix and the linearly structured matrix. Based on the joint distribution of the two test statistics, we derive the asymptotic null distribution and conduct a power analysis for this novel test under the high-dimensional setting. As evidenced by our extensive simulation studies, the proposed integrated test exhibits favorable control of the type I error rate under the null hypothesis, and demonstrates robust power across a spectrum of dense alternative hypotheses. Additionally, we further consider a power-enhanced test statistic that combines the proposed integrated test with a maximum-type test designed for sparse signals, enabling hypothesis testing against both dense and sparse alternatives.