Abstract
In this article, we focus on the problem of testing the equality of several high dimensional mean vectors with unequal covariance matrices. This is one of the most important problems in multivariate statistical analysis and there have been various tests proposed in the literature. Motivated by Bai and Saranadasa (Stat Sin 6:311–329, 1996) and Chen and Qin (Ann Stat 38:808–835, 2010), we introduce a test statistic and derive the asymptotic distributions under the null and the alternative hypothesis. In addition, it is compared with a test statistic recently proposed by Srivastava and Kubokawa (J Multivar Anal 115:204–216, 2013). It is shown that our test statistic performs better especially in the large dimensional case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bai, Z. D., Saranadasa, H. (1996). Effect of high dimension: By an example of a two sample problem. Statistica Sinica, 6, 311–329.
Bai, Z. D., Jiang, D. D., Yao, J. F., Zheng, S. R. (2009). Corrections to LRT on large-dimensional covariance matrix by RMT. The Annals of Statistics, 37(6B), 3822–3840.
Cai, T. T., Xia, Y. (2014). High-dimensional sparse MANOVA. Journal of Multivariate Analysis, 131, 174–196.
Chen, S. X., Qin, Y. L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. The Annals of Statistics, 38(2), 808–835.
Dempster, A. P. (1958). A high dimensional two sample significance test. The Annals of Mathematical Statistics, 29(1), 995–1010.
Dempster, A. P. (1960). A significance test for the separation of two highly multivariate small samples. Biometrics, 16(1), 41.
Fujikoshi, Y., Himeno, T., Wakaki, H. (2004). Asymptotic results of a high dimensional MANOVA test and power comparison when the dimension is large compared to the sample size. Journal of the Japan Statistical Society, 34(1), 19–26.
Fujikoshi, Y., Ulyanov, V. V., Shimizu, R. (2010). Multivariate statistics: High-dimensional and large-sample approximations. Hoboken, New Jersey: Wiley.
Hall, P., Heyde, C. C. (1980). Martingale limit theory and its application. New York: Academic press.
Hotelling, H. (1947). Multivariate quality control, illustrated by the air testing of sample bombsights. In C. Eisenhart, M.W. Hastay, E.A. Wallis (Eds.) Selected Techniques of Statistical Analysis (pp. 111–184). New York: McGraw-Hill.
Jiang, D., Jiang, T., Yang, F. (2012). Likelihood ratio tests for covariance matrices of high-dimensional normal distributions. Journal of Statistical Planning and Inference, 142(8), 2241–2256.
Jiang, T., Yang, F. (2013). Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions. The Annals of Statistics, 41(4), 2029–2074.
Lawley, D. N. (1938). A generalization of Fisher’s z test. Biometrika, 30(1/2), 180–187.
Li, J., Chen, S. X. (2012). Two sample tests for high-dimensional covariance matrices. The Annals of Statistics, 40(2), 908–940.
Pillai, K. C. S. (1955). Some new test criteria in multivariate analysis. The Annals of Mathematical Statistics, 26(1), 117–121.
Schott, J. R. (2007). Some high-dimensional tests for a one-way MANOVA. Journal of Multivariate Analysis, 98(9), 1825–1839.
Srivastava, M. S. (2007). Multivariate theory for analyzing high dimensional data. Journal of the Japan Statistical Society, 37(1), 53–86.
Srivastava, M. S. (2009). A test for the mean vector with fewer observations than the dimension under non-normality. Journal of Multivariate Analysis, 100(3), 518–532.
Srivastava, M. S., Du, M. (2008). A test for the mean vector with fewer observations than the dimension. Journal of Multivariate Analysis, 99(3), 386–402.
Srivastava, M. S., Fujikoshi, Y. (2006). Multivariate analysis of variance with fewer observations than the dimension. Journal of Multivariate Analysis, 97(9), 1927–1940.
Srivastava, M. S., Kubokawa, T. (2013). Tests for multivariate analysis of variance in high dimension under non-normality. Journal of Multivariate Analysis, 115, 204–216.
Srivastava, M. S., Yanagihara, H. (2010). Testing the equality of several covariance matrices with fewer observations than the dimension. Journal of Multivariate Analysis, 101(6), 1319–1329.
Srivastava, M. S., Kollo, T., von Rosen, D. (2011). Some tests for the covariance matrix with fewer observations than the dimension under non-normality. Journal of Multivariate Analysis, 102(6), 1090–1103.
Srivastava, M. S., Katayama, S., Kano, Y. (2013). A two sample test in high dimensional data. Journal of Multivariate Analysis, 114, 349–358.
Wilks, S. S. (1932). Certain generalizations in the analysis of variance. Biometrika, 24(3), 471–494.
Acknowledgments
The authors would like to thank the referees for their constructive comments that led to a substantial improvement of the paper. J. Hu was partially supported by CNSF 11301063. Z. D. Bai was partially supported by CNSF 11171057 and 11571067.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Hu, J., Bai, Z., Wang, C. et al. On testing the equality of high dimensional mean vectors with unequal covariance matrices. Ann Inst Stat Math 69, 365–387 (2017). https://doi.org/10.1007/s10463-015-0543-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-015-0543-8