Abstract
A test statistic for homogeneity of two or more covariance matrices of large dimensions is presented when the data are multivariate normal. The statistic is location-invariant and defined as a function of U-statistics of non-degenerate kernels so that the corresponding asymptotic theory is employed to derive the limiting normal distribution of the test under a few mild and practical assumptions. Accuracy of the test is shown through simulations with different parameter settings.
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References
Ahmad, M. R. 2014. Location-invariant and non-invariant tests for high-dimensional covariance matrices under normality and non-normality. Technical report, Uppsala University, Uppsala, Sweden.
Ahmad, M. R. 2015. U-tests of general linear hypotheses for high-dimensional data under nonnormality and heteroscedasticity. Journal of Statistical Theory & Practice 90 (3):544–70.
Ahmad, M. R. 2017. Location-invariant multi-sample U-tests for covariance matrices with large dimension. Scandinavian Journal of Statistics, in press. doi:https://doi.org/10.1111/sjos.12262.
Aoshima, M., and K. Yata. 2016. Two-sample tests for high-dimension, strongly spiked eigenvalue models. arXiv 1602.02491.
Gretton, A., K. M. Borgwardt, M. J. Rasch, et al. 2008. A kernel method for the two-sample problem. Journal of Machine Learning Research 1:1–43.
Himeno, T., and T. Yamada. 2014. Estimations for some functions of covariance matrix in high dimension under non-normality and its applications. Journal of Multivariate Analysis 130:27–44.
Hoeffding, W. 1948. A class of statistics with asymptotically normal distribution. Annals of Mathematical Statistics 19:293–325.
Jiang, J. 2010. Large sample techniques for statistics. New York, NY: Springer.
Jung, S., and J. S. Marron. 2009. PCA consistency in high dimension, low sample size context. Annals of Statistics 37:4104–30.
Koroljuk, V. S., and Y. V. Borovskich. 1994. Theory of U-statistics. Dordrecht, The Netherlands: Kluwer Academic Press.
Lee, A. J. 1990. U-statistics: Theory and Practice. Boca Raton, FL: CRC Press.
Lehmann, E. L. 1999. Elements of large-sample theory. New York, NY: Springer.
Mathai, A. M., S. B. Provost, and T. Hayakawa. 1995. Bilinear forms and zonal polynomials. Lecture Notes in Statistics, no. 102. New York, NY: Springer.
Muirhead, R. J. 2005. Aspects of multivariate statistical theory. New York, NY: Wiley.
Polansky, A. M. 2011. Introduction to statistical limit theory. Boca Raton, FL: CRC Press.
Schott, J. R. 2007. A test for the equality of covariance matrices when the dimnsion is large relative to the sample size. Computational Statistics & Data Analysis 51:6535–6542.
Searle, S. R. 1971. Linear models. New York, NY: Wiley.
Seber, G. A. F. 2004. Multivariate observations. New York, NY: Wiley.
Serfling, R. J. 1980. Approximation theorems of mathematical statistics. Weinheim, Germany: Wiley.
van der Vaart, A. W. 1998. Asymptotic statistics. New York, NY: Cambridge University Press.
Yata, K., and M. Aoshima 2012. Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations. Journal of Multivariate Analysis 105:193–215.
Yata, K., and M. Aoshima. 2013. PCA consistency for the power spiked model in high-dimensional settings. Journal of Multivariate Analysis 122:334–354.
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The author is thankful to the editor, the associate editor, and two anonymous referees for their comments, which helped improve the original version of the article.
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Ahmad, M.R. Location-invariant tests of homogeneity of large-dimensional covariance matrices. J Stat Theory Pract 11, 731–745 (2017). https://doi.org/10.1080/15598608.2017.1308895
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DOI: https://doi.org/10.1080/15598608.2017.1308895