Robust tests of the equality of two high-dimensional covariance matrices

Abstract

It is of great importance in both theory and application to test the equality of two covariance matrices ${\mathbf{\Sigma }}_1$ and ${\mathbf{\Sigma }}_2$. This article proposes a new robust test based on spatial sign statistic regarding ${\mathrm{H}}_0:{\mathbf{\Sigma }}_1={\mathbf{\Sigma }}_2$ in high-dimensional setting, and shows that the test statistic is asymptotically normal under elliptical distribution. Besides theoretical properties, simulation results also show that the new test significantly outperforms existing methods in terms of size and power for non normal and high-dimensional data. Analysis of colon cancer data set is carried out to demonstrate the application of the testing procedure.

Keywords:

• Asymptotic normality
• High-dimensional data
• Nonparametric method
• Spatial sign
• Robustness

Acknowledgment

The authors are grateful to the referees, Associate Editor, and Editor for their insightful comments that have significantly improved the article.

Additional information

Funding

This research was supported by NNSF of China, [Grants No. 11771332 and 11771220], and NSF of Tianjin [Grant 18JCJQJC46000].