A Robust high dimensional correlation matrices statistic for covariance singularity with outlier

High-dimensional data analysis often challenges classical statistical methods due to covariance matrix singularity and sensitivity to outliers. Jennrich’s statistic is a classical method for testing the equality of correlation matrices. However, its assumptions of normality and well-conditioned correlation matrices can be problematic in high-dimensional data settings. These conditions leading to covariance singularity which can affect the reliability of statistical results. This study addresses these issues by proposing seven robust high-dimensional correlation tests that integrate regularization techniques and robust correlation into Jennrich’s statistic, namely, jb-statistic, jt-statistic, jh-statistic, jmads-statistic, jbmads-statistic, jtmads-statistic, and jhmads-statistic. The performance of these proposed statistics is evaluated through simulation studies by examining the Type I error and power of the tests under various conditions, including different levels of dimensionality, sample sizes, correlation shift, and data contamination by outliers. Furthermore, real-world financial data validates the seven statistics, specifically assessing the correlation structure of currency exchange rates before and after the USA subprime financial crisis. Results showed that the proposed statistics outperform the classical Jennrich’s statistic, especially in terms of maintaining an acceptable Type I error rate and power of test values under various conditions. jmads-statistic, jbmads-statistic, jtmads-statistic, and jhmads-statistic consistently show better control over the Type I error rate in the presence of outliers. The analysis of the power of test values indicates that the jt- statistic and jb-statistic are the most reliable and effective tests across different dimensions, sample sizes, and correlation shifts. Additionally, real application on most currency structures during the USA subprime financial crisis validated that jbmads- statistic is more robust. In conclusion, this study contributes to robust high-dimensional statistical testing by offering enhanced methods capable of handling outliers and singular matrices.