Structural Testing of High-dimensional Correlation Matrices

Due to scale invariance, correlation matrices play a critical role in multivariate statistical analysis.Statistical inference about correlation matrices encounter enormous challenges and is fundamentally different from inference about covariance matrices in both low-and high-dimensional settings.This paper studies the test of general linear structures of high-dimensional correlation matrices, which include commonly-used banded matrices and compound symmetry matrices as special cases.We first propose a procedure using the quadratic loss function to estimate the unknown parameters associated with the assumed linear structure.We then develop test statistics, based on the quadratic and infinite norms, which are suitable for dense and sparse alternatives, respectively.The limiting distributions of our proposed test statistics are derived under the null and alternative hypotheses.Extensive simulation studies are conducted to demonstrate the finite sample performance of our proposed tests.Moreover, a real data example is provided to show the applicability and the practical utility of the tests.