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A high-dimensional test for the equality of the smallest eigenvalues of a covariance matrix

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  • Schott, James R.

Abstract

For the test of sphericity, Ledoit and Wolf [Ann. Statist. 30 (2002) 1081-1102] proposed a statistic which is robust against high dimensionality. In this paper, we consider a natural generalization of their statistic for the test that the smallest eigenvalues of a covariance matrix are equal. Some inequalities are obtained for sums of eigenvalues and sums of squared eigenvalues. These bounds permit us to obtain the asymptotic null distribution of our statistic, as the dimensionality and sample size go to infinity together, by using distributional results obtained by Ledoit and Wolf [Ann. Statist. 30 (2002) 1081-1102]. Some empirical results comparing our test with the likelihood ratio test are also given.

Suggested Citation

  • Schott, James R., 2006. "A high-dimensional test for the equality of the smallest eigenvalues of a covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 97(4), pages 827-843, April.
    • Handle: RePEc:eee:jmvana:v:97:y:2006:i:4:p:827-843
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    Cited by:

    1. Wang, Cheng & Yang, Jing & Miao, Baiqi & Cao, Longbing, 2013. "Identity tests for high dimensional data using RMT," Journal of Multivariate Analysis, Elsevier, vol. 118(C), pages 128-137.
    2. Wagner Martin & Hlouskova Jaroslava, 2015. "Growth Regressions, Principal Components Augmented Regressions and Frequentist Model Averaging," Journal of Economics and Statistics (Jahrbuecher fuer Nationaloekonomie und Statistik), De Gruyter, vol. 235(6), pages 642-662, December.
    3. Wang, Cheng, 2014. "Asymptotic power of likelihood ratio tests for high dimensional data," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 184-189.
    4. Marc Hallin & Marcelo Moreira J. & Alexei Onatski, 2013. "Group Invariance, Likelihood Ratio Tests, and the Incidental Parameter Problem in a High-Dimensional Linear Model," Working Papers ECARES ECARES 2013-04, ULB -- Universite Libre de Bruxelles.
    5. Virta, Joni, 2021. "Testing for subsphericity when n and p are of different asymptotic order," Statistics & Probability Letters, Elsevier, vol. 179(C).
    6. Joni Virta & Niko Lietzén & Henri Nyberg, 2024. "Robust signal dimension estimation via SURE," Statistical Papers, Springer, vol. 65(5), pages 3007-3038, July.
    7. Fisher, Thomas J. & Sun, Xiaoqian & Gallagher, Colin M., 2010. "A new test for sphericity of the covariance matrix for high dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2554-2570, November.
    8. Nordhausen, Klaus & Oja, Hannu & Tyler, David E., 2022. "Asymptotic and bootstrap tests for subspace dimension," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    9. Schott, James R., 2007. "A test for the equality of covariance matrices when the dimension is large relative to the sample sizes," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6535-6542, August.
    10. Jaroslava Hlouskova & Martin Wagner, 2013. "The Determinants of Long-Run Economic Growth: A Conceptually and Computationally Simple Approach," Swiss Journal of Economics and Statistics (SJES), Swiss Society of Economics and Statistics (SSES), vol. 149(IV), pages 445-492, December.
    11. Wagner, Martin & Hlouskova, Jaroslava, 2009. "Growth Regressions, Principal Components and Frequentist Model Averaging," Economics Series 236, Institute for Advanced Studies.
    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. Bernard, Gaspard & Verdebout, Thomas, 2024. "On testing the equality of latent roots of scatter matrices under ellipticity," Journal of Multivariate Analysis, Elsevier, vol. 199(C).
    14. Alexei Onatski & Marcelo Moreira J. & Marc Hallin, 2012. "Signal Detection in High Dmension: The Multispiked Case," Working Papers ECARES ECARES 2012-036, ULB -- Universite Libre de Bruxelles.
    15. Elisabeta JABA & Alina Mariuca IONESCU & Corneliu IATU & Christiana Brigitte BALAN, 2009. "The Evaluation Of The Regional Profile Of The Economic Development In Romania," Analele Stiintifice ale Universitatii "Alexandru Ioan Cuza" din Iasi - Stiinte Economice (1954-2015), Alexandru Ioan Cuza University, Faculty of Economics and Business Administration, vol. 56, pages 537-549, November.
    16. Fujikoshi, Yasunori & Yamada, Takayuki & Watanabe, Daisuke & Takakazu Sugiyama, 2007. "Asymptotic distribution of the LR statistic for equality of the smallest eigenvalues in high-dimensional principal component analysis," Journal of Multivariate Analysis, Elsevier, vol. 98(10), pages 2002-2008, November.

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