For high-dimensional small sample size data, Hotelling’s T 2 test is not applicable for testing mean vectors due to the singularity problem in the sample covariance matrix. To overcome the problem, there are three main approaches in the literature. Note, however, that each of the existing approaches may have serious limitations and only works well in certain situations. Inspired by this, we propose a pairwise Hotelling method for testing high-dimensional mean vectors, which, in essence, provides a good balance between the existing approaches. To effectively utilize the correlation information, we construct the new test statistics as the summation of Hotelling’s test statistics for the covariate pairs with strong correlations and the squared t statistics for the individual covariates that have little correlation with others. We further derive the asymptotic null distributions and power functions for the proposed Hotelling’s tests under some regularity conditions. Numerical results show that our new tests are able to control the type I error rates, and can achieve a higher statistical power compared to existing methods, especially when the covariates are highly correlated. Two real data examples are also analyzed and they both demonstrate the efficacy of our pairwise Hotelling’s tests.
Bibliographical noteKAUST Repository Item: Exported on 2022-07-05
Acknowledgements: The authors thank the Editor, Associate Editor, and two anonymous reviewers for their insightful comments and suggestions, which led to a significant improvement of this article. Zongliang Hu’s research was supported by the National Natural Science Foundation of China (12001378), the Guangdong Basic and Applied Basic Research Foundation (2019A1515110449), and the Natural Science Foundation of Gangdong Province (2020A1515010372). Tiejun Tong’s research was supported by the General Research Funds (HKBU12303918, HKBU12303421), the Initiation Grant for Faculty Niche Research Areas (RC-FNRA-IG/20-21/SCI/03) of Hong Kong Baptist University, and the National Natural Science Foundation of China (1207010822). Marc G. Genton’s research was supported by the King Abdullah University of Science and Technology (KAUST).
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty