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Testing for independence of large dimensional vectors

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  • Bodnar, Taras
  • Dette, Holger
  • Parolya, Nestor

Abstract

In this paper, new tests for the independence of two high-dimensional vectors are investigated. We consider the case where the dimension of the vectors increases with the sample size and propose multivariate analysis of variance-type statistics for the hypothesis of a block diagonal covariance matrix. The asymptotic properties of the new test statistics are investigated under the null hypothesis and the alternative hypothesis using random matrix theory. For this purpose, we study the weak convergence of linear spectral statistics of central and (conditionally) noncentral Fisher matrices. In particular, a central limit theorem for linear spectral statistics of large dimensional(conditionally) noncentral Fisher matrices is derived which is then used to analyse the power of the tests under the alternative. The theoretical results are illustrated by means of a simulation study where we also compare the new tests with several alternatives, in particular with the commonly used corrected likelihood ratio test. It is demonstrated that the latter test does not keep its nominal level if the dimension of one sub-vector is relatively small compared to the dimension of the other sub-vector.On the other hand, the tests proposed in this paper provide a reasonable approximation of the nominal level in such situations. Moreover, we observe that one of the proposed tests is most powerful under a variety of correlation scenarios.

Suggested Citation

  • Bodnar, Taras & Dette, Holger & Parolya, Nestor, 2019. "Testing for independence of large dimensional vectors," MPRA Paper 97997, University Library of Munich, Germany, revised May 2019.
    • Handle: RePEc:pra:mprapa:97997
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    References listed on IDEAS

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    2. Emilie Devijver & Mélina Gallopin, 2018. "Block-Diagonal Covariance Selection for High-Dimensional Gaussian Graphical Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(521), pages 306-314, January.
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    5. Masashi Hyodo & Nobumichi Shutoh & Takahiro Nishiyama & Tatjana Pavlenko, 2015. "Testing block-diagonal covariance structure for high-dimensional data," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 69(4), pages 460-482, November.
    6. Yamada, Yuki & Hyodo, Masashi & Nishiyama, Takahiro, 2017. "Testing block-diagonal covariance structure for high-dimensional data under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 305-316.
    7. G. Pan & J. Gao & Y. Yang & M. Guo, 2012. "Independence Test for High Dimensional Random Vectors," Monash Econometrics and Business Statistics Working Papers 1/12, Monash University, Department of Econometrics and Business Statistics.
    8. Bodnar, Taras & Gupta, Arjun K. & Parolya, Nestor, 2014. "On the strong convergence of the optimal linear shrinkage estimator for large dimensional covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 132(C), pages 215-228.
    9. Birke, Melanie & Dette, Holger, 2005. "A note on testing the covariance matrix for large dimension," Statistics & Probability Letters, Elsevier, vol. 74(3), pages 281-289, October.
    10. Dozier, R. Brent & Silverstein, Jack W., 2007. "On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices," Journal of Multivariate Analysis, Elsevier, vol. 98(4), pages 678-694, April.
    11. Kakizawa, Yoshihide & Iwashita, Toshiya, 2008. "A comparison of higher-order local powers of a class of one-way MANOVA tests under general distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1128-1153, July.
    12. Schott, James R., 2007. "Some high-dimensional tests for a one-way MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1825-1839, October.
    13. Arjun Gupta & Jin Xu, 2006. "On Some Tests of the Covariance Matrix Under General Conditions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(1), pages 101-114, March.
    14. Cai, Tony & Liu, Weidong & Luo, Xi, 2011. "A Constrained â„“1 Minimization Approach to Sparse Precision Matrix Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 594-607.
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    Citations

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    Cited by:

    1. Taras Bodnar & Holger Dette & Nestor Parolya & Erik Thors'en, 2019. "Sampling Distributions of Optimal Portfolio Weights and Characteristics in Low and Large Dimensions," Papers 1908.04243, arXiv.org, revised Apr 2023.
    2. Xu, Kai & Hao, Xinxin, 2019. "A nonparametric test for block-diagonal covariance structure in high dimension and small samples," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 551-567.
    3. Bodnar, Taras & Parolya, Nestor & Thorsén, Erik, 2023. "Is the empirical out-of-sample variance an informative risk measure for the high-dimensional portfolios?," Finance Research Letters, Elsevier, vol. 54(C).
    4. Bauder, David & Bodnar, Taras & Parolya, Nestor & Schmid, Wolfgang, 2020. "Bayesian inference of the multi-period optimal portfolio for an exponential utility," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    5. Bodnar, Olha & Touli, Elena Farahbakhsh, 2023. "Exact test theory in Gaussian graphical models," Journal of Multivariate Analysis, Elsevier, vol. 196(C).
    6. Jiayu Lai & Xiaoyi Wang & Kaige Zhao & Shurong Zheng, 2023. "Block-diagonal test for high-dimensional covariance matrices," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(1), pages 447-466, March.
    7. Mingyue Hu & Yongcheng Qi, 2023. "Limiting distributions of the likelihood ratio test statistics for independence of normal random vectors," Statistical Papers, Springer, vol. 64(3), pages 923-954, June.
    8. Dette, Holger & Dörnemann, Nina, 2020. "Likelihood ratio tests for many groups in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
    9. Taras Bodnar & Solomiia Dmytriv & Yarema Okhrin & Nestor Parolya & Wolfgang Schmid, 2020. "Statistical inference for the EU portfolio in high dimensions," Papers 2005.04761, arXiv.org.
    10. Feng, Long & Zhang, Xiaoxu & Liu, Binghui, 2020. "Multivariate tests of independence and their application in correlation analysis between financial markets," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    11. Bodnar, Olha & Bodnar, Taras & Parolya, Nestor, 2022. "Recent advances in shrinkage-based high-dimensional inference," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    12. Dörnemann, Nina, 2023. "Likelihood ratio tests under model misspecification in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 193(C).

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    More about this item

    Keywords

    Testing for independence; large dimensional covariance matrix; noncentral Fisher random matrix; linear spectral statistics; asymptotic normality;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General

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