Global one-sample tests for high-dimensional covariance matrices

Abstract

Testing high-dimensional covariance matrix plays an important role in multivariate statistical analysis. Many statisticians used the statistics based on $\mathrm{t}\mathrm{r}\left[\right(\mathbf{\Sigma }{\mathbf{\Sigma }}_0^{-1}-{\mathit{I}}_p{)}^2]$ to test $H_0:\mathbf{\Sigma }={\mathbf{\Sigma }}_0$ with $\mathbf{\Sigma }$ being the covariance matrix. However, none have proposed a statistic based on $\mathrm{t}\mathrm{r}\left[\right(\mathbf{\Sigma }-{\mathbf{\Sigma }}_0{)}^2]$ for this purpose. In fact, neither of the two tests is superior to the other based on their powers because they target different kinds of dense alternatives. Furthermore, some maximum-type tests were proposed to accommodate sparse alternatives. By using the advantages of these tests, we propose two new tests for one-sample covariance matrix when the sample size and data dimension increase proportionally. One is suitable for dense alternatives, the other is powerful against a wide range of situations, such as dense, sparse or a mixture of both alternatives. Extensive simulation results show that our proposed tests maintain high powers against various alternatives while the existing tests fail in at least one situation.

Keywords:

• Dense alternative
• high dimension
• random matrix theory
• sparse alternative

The authors would like to thank the associate editor, the anonymous referees and the editor for their insightful comments and suggestions which substantially improve the article. Zheng's research was supported by NSFC grants 12071066 and 11690012, and Liu's research was supported by the Liaoning Provincial Education Department, China (No. LN2020J18).

Disclosure statement

No potential conflict of interest was reported by the author(s).