A high-dimensional test for the k-sample Behrens–Fisher problem

Abstract

In this paper, the problem of testing the equality of the mean vectors of k populations with possibly unknown and unequal covariance matrices is investigated in high-dimensional settings. The null distributions of most existing tests are asymptotically normal which inevitably imposes strong conditions on covariance matrices. However, we assume here only mild additional conditions on the proposed test, which offers much flexibility in practical applications. Additionally, the Welch–Satterthwaite ${\chi }^2$-approximation we adopted can automatically mimic the shape of the null distribution of the proposed test statistic, while the normal approximation cannot achieve the adaptivity. Finally, an extensive simulation study shows that the proposed test has better performance on both size and power compared with existing methods.

Keywords:

• High dimension
• Behrens–Fisher problem
• heteroscedasticity
• Welch–Satterthwaite ${\chi }^2$-approximation

Acknowledgements

The authors would like to thank the Editor, the Associate Editor and the anonymous Reviewer for their valuable comments and suggestions on earlier versions of this paper. The authors are grateful to Dr Bu Zhou from School of Statistics and Mathematics, Zhejiang Gongshang University for sharing the cornea surface data with us.

Disclosure statement

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

Additional information

Funding

He's work is supported by National Natural Science Foundation of China [grant number 11201005], Anhui Provincial Natural Science Foundation [grant number 2008085MA08] and Foundation of Anhui Provincial Education Department [grant number KJ2021A1523]. Xu's work is supported by National Natural Science Foundation of China [grant numbers 12271005, 11901006] and Natural Science Foundation of Anhui Province [grant number 1908085QA06]. Cao's work is supported by the National Natural Science Foundation of China [grant number 11601008], Humanities and Social Sciences Foundation of Ministry of Education, China [grant number 22YJC910001] and Anhui Provincial Natural Science Foundation [grant number 2108085MA09].