Skip to main content
SpringerLink
Log in

Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

Consider a p-variate normal random vector. We are interested in the limiting distributions of likelihood ratio test (LRT) statistics for testing the independence of its grouped components based on a random sample of size n. In classical multivariate analysis, the dimension p is fixed or relatively small, and the limiting distribution of the LRT is a chi-square distribution. When p goes to infinity, the chi-square approximation to the classical LRT statistic may be invalid. In this paper, we prove that the LRT statistic converges to a normal distribution under quite general conditions when p goes to infinity. We propose an adjusted test statistic which has a chi-square limit in general. Our comparison study indicates that the adjusted test statistic outperforms among the three approximations in terms of sizes. We also report some numerical results to compare the performance of our approaches and other methods in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Ahlfors, L. V. (1979). Complex analysis: An introduction to the theory of analytic functions of one complex variable (3rd ed.). New York: McGraw-Hill.

    MATH  Google Scholar 

  • Bai, Z., Jiang, D., Yao, J., Zheng, S. (2009). Corrections to LRT on large dimensional covariance matrix by RMT. Annals of Statistics, 37(6B), 3822–3840.

  • Bao, Z., Hu, J., Pan, G., Zhou, W. (2017). Test of independence for high-dimensional random vectors based on freeness in block correlation matrices. Electronic Journal of Statistics, 11(1), 1527–1548.

  • Chen, S. X., Zhang, L. X., Zhong, P. S. (2010). Tests for high-dimensional covariance matrices. Journal of the American Statistical Association, 105(490), 810–819.

  • 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, D., Bai, Z., Zheng, S. (2013). Testing the independence of sets of large-dimensional variables. Science China Mathematics, 56(1), 135–147.

  • Jiang, T., Qi, Y. (2015a). Likelihood ratio tests for high-dimensional normal distributions. Scandinavian Journal of Statistics, 42(4), 988–1009.

  • Jiang, T., Qi, Y. (2015b). Supplement to “Likelihood ratio tests for high-dimensional normal Distributions”. http://www.stat.umn.edu/~tjiang/papers/SJSJQ.pdf.

  • Jiang, T., Yang, F. (2013). Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions. Annals of Statistics, 41(4), 2029–2074.

  • 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(4), 1081–1102.

  • Li, W., Chen, J., Yao, J. (2017). Testing the independence of two random vectors where only one dimension is large. Statistics, 51(1), 141–153.

  • Muirhead, R. J. (1982). Aspects of multivariate statistical theory. New York: Wiley.

    Book  MATH  Google Scholar 

  • Schott, J. R. (2001). Some tests for the equality of covariance matrices. Journal of Statistical Planning and Inference, 94(1), 25–36.

    Article  MathSciNet  MATH  Google Scholar 

  • Schott, J. R. (2005). Testing for complete independence in high dimensions. Biometrika, 92(4), 951–956.

    Article  MathSciNet  MATH  Google Scholar 

  • Schott, J. R. (2007). A test for the equality of covariance matrices when the dimension is large relative to the sample sizes. Computational Statistics and Data Analysis, 51(12), 6535–6542.

    Article  MathSciNet  MATH  Google Scholar 

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

  • Wilks, S. S. (1935). On the independence of k sets of normally distributed statistical variables. Econometrica, 3(3), 309–326.

    Article  MATH  Google Scholar 

  • Young, R. M. (1991). 75.9 Euler’s Constant. Mathematical Gazette, 75(472), 187–190.

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank three anonymous reviewers for their constructive comments and suggestions that have led to improvements in the paper. Qi’s research was supported by NSF Grant DMS-1005345, and Wang’s research was supported by NSFC Grant No. 11671021, NSFC Grant No. 11471222 and Foundation of Beijing Education Bureau Grant No. 201510028002.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Minnesota Duluth, 1117 University Drive, Duluth, MN, 55812, USA

    Yongcheng Qi

  2. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Fang Wang

  3. School of Statistics, University of Minnesota, 224 Church Street S. E., Minneapolis, MN, 55455, USA

    Lin Zhang

Authors
  1. Yongcheng Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fang Wang.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qi, Y., Wang, F. & Zhang, L. Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors. Ann Inst Stat Math 71, 911–946 (2019). https://doi.org/10.1007/s10463-018-0666-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-018-0666-9

Keywords

Search

Navigation