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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.
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    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 97 (2006)
    Issue (Month): 4 (April)
    Pages: 827-843

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    Handle: RePEc:eee:jmvana:v:97:y:2006:i:4:p:827-843

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    Keywords: Principal components analysis Sums of eigenvalues;

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    Cited by:
    1. 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.
    2. 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, Alexandru Ioan Cuza University, Faculty of Economics and Business Administration, vol. 56, pages 537-549, November.
    3. 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.
    4. Wang, Cheng, 2014. "Asymptotic power of likelihood ratio tests for high dimensional data," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 184-189.
    5. Wagner, Martin & Hlouskova, Jaroslava, 2009. "Growth Regressions, Principal Components and Frequentist Model Averaging," Economics Series 236, Institute for Advanced Studies.
    6. 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.
    7. 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.
    8. 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.

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