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The Asymptotic Distribution of Singular-Values with Applications to Canonical Correlations and Correspondence Analysis

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  • Eaton, M. L.
  • Tyler, D.

Abstract

Let Xn, n = 1, 2, ... be a sequence of p - q random matrices, p >= q. Assume that for a fixed p - q matrix B and a sequence of constants bn --> [infinity], the random matrix bn(Xn - B) converges in distribution to Z. Let [psi](Xn) denote the q-vector of singular values of Xn. Under these assumptions, the limiting distribution of bn ([psi](Xn) - [psi](B)) is characterized as a function of B and of the limit matrix Z. Applications to canonical correlations and to correspondence analysis are given.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:50:y:1994:i:2:p:238-264
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    Cited by:

    1. Marcos Rangel & Duncan Thomas, 2019. "Decision-Making in Complex Households," NBER Working Papers 26511, National Bureau of Economic Research, Inc.
    2. An, Baiguo & Guo, Jianhua & Wang, Hansheng, 2013. "Multivariate regression shrinkage and selection by canonical correlation analysis," Computational Statistics & Data Analysis, Elsevier, vol. 62(C), pages 93-107.
    3. Charles Lindsey & Simon Sheather & Joseph McKean, 2014. "Using sliced mean variance–covariance inverse regression for classification and dimension reduction," Computational Statistics, Springer, vol. 29(3), pages 769-798, June.
    4. Ogasawara, Haruhiko, 2007. "Asymptotic expansions of the distributions of estimators in canonical correlation analysis under nonnormality," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1726-1750, October.
    5. Thomas, Duncan & Rangel, Marcos, 2020. "Decision-Making in Complex Households," CEPR Discussion Papers 14278, C.E.P.R. Discussion Papers.
    6. Haruhiko Ogasawara, 2009. "Asymptotic expansions in the singular value decomposition for cross covariance and correlation under nonnormality," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(4), pages 995-1017, December.
    7. 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.
    8. F. Chiaromonte, 1998. "On Multivariate Structures and Exhaustive Reductions," Working Papers ir98080, International Institute for Applied Systems Analysis.
    9. Taskinen, Sara & Croux, Christophe & Kankainen, Annaliisa & Ollila, Esa & Oja, Hannu, 2006. "Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 359-384, February.
    10. F. Chiaromonte, 1997. "A Reduction Paradigm for Multivariate Laws," Working Papers ir97015, International Institute for Applied Systems Analysis.
    11. Yin, Xiangrong & Dennis Cook, R., 2004. "Asymptotic distribution of test statistic for the covariance dimension reduction methods in regression," Statistics & Probability Letters, Elsevier, vol. 68(4), pages 421-427, July.
    12. Bai, Z. D. & He, Xuming, 2004. "A chi-square test for dimensionality with non-Gaussian data," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 109-117, January.
    13. Eliana Christou, 2020. "Robust dimension reduction using sliced inverse median regression," Statistical Papers, Springer, vol. 61(5), pages 1799-1818, October.
    14. Marcos A. Rangel & Duncan Thomas, 2019. "Decision-Making in Complex Households," Working Papers 2019-070, Human Capital and Economic Opportunity Working Group.
    15. Liu, Xuejing & Huo, Lei & Wen, Xuerong Meggie & Paige, Robert, 2017. "A link-free approach for testing common indices for three or more multi-index models," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 236-245.
    16. Nordhausen, Klaus & Oja, Hannu & Tyler, David E., 2022. "Asymptotic and bootstrap tests for subspace dimension," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    17. Dümbgen, Lutz, 1995. "A simple proof and refinement of Wielandt's eigenvalue inequality," Statistics & Probability Letters, Elsevier, vol. 25(2), pages 113-115, November.
    18. Yamada, Tomoya, 2013. "Asymptotic properties of canonical correlation analysis for one group with additional observations," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 389-401.
    19. Yuan, Ke-Hai & Bentler, Peter M., 2000. "Inferences on Correlation Coefficients in Some Classes of Nonnormal Distributions," Journal of Multivariate Analysis, Elsevier, vol. 72(2), pages 230-248, February.
    20. Kuriki, Satoshi, 2005. "Asymptotic distribution of inequality-restricted canonical correlation with application to tests for independence in ordered contingency tables," Journal of Multivariate Analysis, Elsevier, vol. 94(2), pages 420-449, June.
    21. Bura, Efstathia & Cook, R. Dennis, 2003. "Rank estimation in reduced-rank regression," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 159-176, October.
    22. Cook, R. Dennis & Yin, Xiangrong, 2002. "Asymptotic distributions for testing dimensionality in q-based pHd," Statistics & Probability Letters, Elsevier, vol. 58(3), pages 233-243, July.
    23. Boik, Robert J., 1998. "A Local Parameterization of Orthogonal and Semi-Orthogonal Matrices with Applications," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 244-276, November.
    24. Fan, Jianqing & Fan, Yingying & Lv, Jinchi, 2008. "High dimensional covariance matrix estimation using a factor model," Journal of Econometrics, Elsevier, vol. 147(1), pages 186-197, November.

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