Performance Evaluation of Two Tests for the Equality of High-Dimensional Mean Vectors with Unequal Covariance Matrices

Abstract

The rapid development of measurement technology in fields such as genetics, medicine, and economics has made high-dimensional data analysis an essential practice, as analysts and researchers need to collect and analyze data from a large number of variables. Analyzing high-dimensional data requires advanced statistical techniques as conventional methods like comparing mean vectors using Hotelling's T²-test are not applicable. In this study, we compare the performance of two tests, T_MCQ and T_SKK, for testing the equality of high-dimensional mean vectors when population covariance matrices are not equal. The T_MCQ was initially proposed by Chen, S. X. and Qin, Y. L. in 2010 and modified by Srivastava, M. S., Katayama, S., & Kano, Y. in 2013, while the T_SKK was developed by Srivastava, M. S. et al. in 2013. A simulation study was conducted using two independent normal samples of equal size. The population covariance matrices were created under five structures: Sphericity, Compound Symmetry (CS), Heterogeneous Compound Symmetry (CSH), Toeplitz, and Block Diagonal (BD) matrix. The results showed that the choice of covariance structure had an impact on the performance of both tests. The T_MCQ test performed well for certain covariance structures including Sphericity, Toeplitz, and BD, while the T_SKK performed well for the CS covariance structure. The performance of T_SKK increased for large sample sizes of at least 60 under the covariance structures of Toeplitz and BD. Under the covariance structures of Sphericity, Toeplitz, and BD, the T_MCQ outperformed the T_SKK, while T_SKK was more efficient than T_MCQ when the covariance matrix structure was CS. Additionally, the performance of T_SKK improved when the sample size was at least 60. We also examined the performance of the two tests when the difference between two population covariance matrices increased and found that both tests continued to perform well for certain covariance structures. However, under the covariance structure of CSH, both tests underperformed for the case being studied