GLHT for high-dimensional covariance matrices: A normal-reference approach*

Statistical inference for high-dimensional data is increasingly essential due to the prevalence of big data. While recent advancements have improved testing linear hypotheses for high-dimensional mean vectors, exploration of testing linear hypotheses for high-dimensional covariance matrices remains limited. This paper focuses on a general linear hypothesis testing (GLHT) problem for high-dimensional covariance matrices, encompassing various specific cases such as testing the equality of variances, testing for a given covariance matrix, assessing the homogeneity of covariance matrices, and examining linear combinations of covariance matrices. We propose and study a new test statistic for the GLHT problem on high-dimensional covariance matrices. Under certain regularity conditions and the null hypothesis, we demonstrate that the test statistic shares the same limiting distribution as a chi-squared-type mixture. This mixture’s distribution can be accurately approximated using a three-cumulant matched chi-squared-approximation approach. Additionally, we establish the asymptotic power of the proposed test against some local alternatives. Simulation studies and a real-world financial data analysis demonstrate that the proposed test outperforms several competitors in terms of size control.