Hypothesis tests for high-dimensional covariance structures

Abstract

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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Acknowledgements

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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Keywords

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