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Asymptotic and bootstrap tests for subspace dimension

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  • Nordhausen, Klaus
  • Oja, Hannu
  • Tyler, David E.

Abstract

Many linear dimension reduction methods proposed in the literature can be formulated using an appropriate pair of scatter matrices. The eigen-decomposition of one scatter matrix with respect to another is then often used to determine the dimension of the signal subspace and to separate signal and noise parts of the data. Three popular dimension reduction methods, namely principal component analysis (PCA), fourth order blind identification (FOBI) and sliced inverse regression (SIR) are considered in detail and the first two moments of subsets of the eigenvalues are used to test for the dimension of the signal space. The limiting null distributions of the test statistics are discussed and novel bootstrap strategies are suggested for the small sample cases. In all three cases, consistent test-based estimates of the signal subspace dimension are introduced as well. The asymptotic and bootstrap tests are illustrated in real data examples.

Suggested Citation

  • Nordhausen, Klaus & Oja, Hannu & Tyler, David E., 2022. "Asymptotic and bootstrap tests for subspace dimension," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    • Handle: RePEc:eee:jmvana:v:188:y:2022:i:c:s0047259x21001081
    DOI: 10.1016/j.jmva.2021.104830
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    References listed on IDEAS

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    1. Thomas P. Hettmansperger, 2002. "A practical affine equivariant multivariate median," Biometrika, Biometrika Trust, vol. 89(4), pages 851-860, December.
    2. Annaliisa Kankainen & Sara Taskinen & Hannu Oja, 2007. "Tests of multinormality based on location vectors and scatter matrices," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 16(3), pages 357-379, November.
    3. Nordhausen, Klaus & Oja, Hannu & Tyler, David E., 2008. "Tools for Exploring Multivariate Data: The Package ICS," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 28(i06).
    4. Dray, Stephane, 2008. "On the number of principal components: A test of dimensionality based on measurements of similarity between matrices," Computational Statistics & Data Analysis, Elsevier, vol. 52(4), pages 2228-2237, January.
    5. Liping Zhu & Tao Wang & Lixing Zhu & Louis Ferré, 2010. "Sufficient dimension reduction through discretization-expectation estimation," Biometrika, Biometrika Trust, vol. 97(2), pages 295-304.
    6. David E. Tyler & Frank Critchley & Lutz Dümbgen & Hannu Oja, 2009. "Invariant co‐ordinate selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 549-592, June.
    7. Ye Z. & Weiss R.E., 2003. "Using the Bootstrap to Select One of a New Class of Dimension Reduction Methods," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 968-979, January.
    8. Eaton, M. L. & Tyler, D., 1994. "The Asymptotic Distribution of Singular-Values with Applications to Canonical Correlations and Correspondence Analysis," Journal of Multivariate Analysis, Elsevier, vol. 50(2), pages 238-264, August.
    9. Zhu, Lixing & Miao, Baiqi & Peng, Heng, 2006. "On Sliced Inverse Regression With High-Dimensional Covariates," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 630-643, June.
    10. Bura, E. & Yang, J., 2011. "Dimension estimation in sufficient dimension reduction: A unifying approach," Journal of Multivariate Analysis, Elsevier, vol. 102(1), pages 130-142, January.
    11. Klaus Nordhausen & David E. Tyler, 2015. "A cautionary note on robust covariance plug-in methods," Biometrika, Biometrika Trust, vol. 102(3), pages 573-588.
    12. Weisberg, Sanford, 2002. "Dimension Reduction Regression in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 7(i01).
    13. 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.
    14. Wei Luo & Bing Li, 2016. "Combining eigenvalues and variation of eigenvectors for order determination," Biometrika, Biometrika Trust, vol. 103(4), pages 875-887.
    15. Yanyuan Ma & Liping Zhu, 2013. "A Review on Dimension Reduction," International Statistical Review, International Statistical Institute, vol. 81(1), pages 134-150, April.
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