Hypothesis tests for high-dimensional covariance structures


We consider hypothesis testing for high-dimensional covariance structures in which the covariance matrix is a (i) scaled identity matrix, (ii) diagonal matrix, or (iii) intraclass covariance matrix. Our purpose is to systematically establish a nonparametric approach for testing the high-dimensional covariance structures (i)–(iii). We produce a new common test statistic for each covariance structure and show that the test statistic is an unbiased estimator of its corresponding test parameter. We prove that the test statistic establishes the asymptotic normality. We propose a new test procedure for (i)–(iii) and evaluate its asymptotic size and power theoretically when both the dimension and sample size increase. We investigate the performance of the proposed test procedure in simulations. As an application of testing the covariance structures, we give a test procedure to identify an eigenvector. Finally, we demonstrate the proposed test procedure by using a microarray data set.

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  1. Alon, U., Barkai, N., Notterman, D. A., Gish, K., Ybarra, S., Mack, D., Levine, A. J. (1999). Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays. Proceedings of the National Academy of Sciences of the United States of America, 96, 6745–6750.

    Article  Google Scholar 

  2. Aoshima, M., Yata, K. (2015). Asymptotic normality for inference on multisample, high-dimensional mean vectors under mild conditions. Methodology and Computing in Applied Probability, 17, 419–439.

    MathSciNet  Article  Google Scholar 

  3. Aoshima, M., Yata, K. (2018). Two-sample tests for high-dimension, strongly spiked eigenvalue models. Statistica Sinica, 28, 43–62.

    MathSciNet  MATH  Google Scholar 

  4. Aoshima, M., Yata, K. (2019). Distance-based classifier by data transformation for high-dimension, strongly spiked eigenvalue models. Annals of the Institute of Statistical Mathematics, 71, 473–503.

    MathSciNet  Article  Google Scholar 

  5. Bai, Z., Saranadasa, H. (1996). Effect of high dimension: By an example of a two sample problem. Statistica Sinica, 6, 311–329.

    MathSciNet  MATH  Google Scholar 

  6. Bao, Z., Lin, L.-C., Pan, G., Zhou, W. (2015). Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application. Annals of Statistics, 43, 2588–2623.

    MathSciNet  MATH  Google Scholar 

  7. Chen, S. X., Qin, Y.-L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Annals of Statistics, 38, 808–835.

    MathSciNet  MATH  Google Scholar 

  8. Chen, S. X., Zhang, L.-X., Zhong, P.-S. (2010). Tests for high-dimensional covariance matrices. Journal of the American Statistical Association, 105, 810–819.

    MathSciNet  Article  Google Scholar 

  9. Ishii, A., Yata, K., Aoshima, M. (2016). Asymptotic properties of the first principal component and equality tests of covariance matrices in high-dimension, low-sample-size context. Journal of Statistical Planning and Inference, 170, 186–199.

    MathSciNet  Article  Google Scholar 

  10. Ishii, A., Yata, K., Aoshima, M. (2019). Equality tests of high-dimensional covariance matrices under the strongly spiked eigenvalue model. Journal of Statistical Planning and Inference, 202, 99–111.

    MathSciNet  Article  Google Scholar 

  11. Ledoit, O., Wolf, M. (2002). Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size. Annals of Statistics, 30, 1081–1102.

    MathSciNet  Article  Google Scholar 

  12. Schott, J. R. (2005). Testing for complete independence in high dimensions. Biometrika, 92, 951–956.

    MathSciNet  Article  Google Scholar 

  13. Srivastava, M. S., Kollo, T., Rosen, V. D. (2011). Some tests for the covariance matrix with fewer observations than the dimension under non-normality. Journal of Multivariate Analysis, 102, 1090–1103.

    MathSciNet  Article  Google Scholar 

  14. Srivastava, M. S., Reid, N. (2012). Testing the structure of the covariance matrix with fewer observations than the dimension. Journal of Multivariate Analysis, 112, 156–171.

    MathSciNet  Article  Google Scholar 

  15. Yata, K., Aoshima, M. (2010). Effective PCA for high-dimension, low-sample-size data with singular value decomposition of cross data matrix. Journal of Multivariate Analysis, 101, 2060–2077.

    MathSciNet  Article  Google Scholar 

  16. Yata, K., Aoshima, M. (2012). Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations. Journal of Multivariate Analysis, 105, 193–215.

    MathSciNet  Article  Google Scholar 

  17. Yata, K., Aoshima, M. (2013). Correlation tests for high-dimensional data using extended cross-data-matrix methodology. Journal of Multivariate Analysis, 117, 313–331.

    MathSciNet  Article  Google Scholar 

  18. Yata, K., Aoshima, M. (2016). High-dimensional inference on covariance structures via the extended cross-data-matrix methodology. Journal of Multivariate Analysis, 151, 151–166.

    MathSciNet  Article  Google Scholar 

  19. Yata, K., Aoshima, M., Nakayama, Y. (2018). A test of sphericity for high-dimensional data and its application for detection of divergently spiked noise. Sequential Analysis, 37, 397–411.

    MathSciNet  Article  Google Scholar 

  20. Zhong, P.-S., Lan, W., Song, X. K. P., Tsai, C. H. (2017). Tests for covariance structures with high-dimensional repeated measurements. Annals of Statistics, 45, 1185–1213.

    MathSciNet  Article  Google Scholar 

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Research of the first author was partially supported by Grant-in-Aid for Young Scientists, Japan Society for the Promotion of Science (JSPS), under Contract Number 18K18015. Research of the second author was partially supported by Grant-in-Aid for Scientific Research (C), JSPS, under Contract Number 18K03409. Research of the third author was partially supported by Grants-in-Aid for Scientific Research (A) and Challenging Research (Exploratory), JSPS, under Contract Numbers 20H00576 and 19K22837.

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Correspondence to Makoto Aoshima.

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Ishii, A., Yata, K. & Aoshima, M. Hypothesis tests for high-dimensional covariance structures. Ann Inst Stat Math 73, 599–622 (2021). https://doi.org/10.1007/s10463-020-00760-5

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  • Cross-data-matrix methodology
  • Diagonal structure
  • Intraclass correlation model
  • Test of eigenvector
  • Unbiased estimate