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Asymptotics of Eigenvalues and Unit-Length Eigenvectors of Sample Variance and Correlation Matrices

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  • Kollo, T.
  • Neudecker, H.

Abstract

Multivariate asymptotic (normal) distributions for eigenvalues and unit-length eigenvectors of sample variance and correlation matrices are derived. Beside the general case, when existence of the (finite) fourth-order moments of the population distribution is assumed, formulae for the asymptotic variance matrices in the cases of normal and elliptical populations are also derived. It is assumed throughout that population variance and correlation matrices are nonsingular and without multiple eigenvalues.

Suggested Citation

  • Kollo, T. & Neudecker, H., 1993. "Asymptotics of Eigenvalues and Unit-Length Eigenvectors of Sample Variance and Correlation Matrices," Journal of Multivariate Analysis, Elsevier, vol. 47(2), pages 283-300, November.
    • Handle: RePEc:eee:jmvana:v:47:y:1993:i:2:p:283-300
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    Cited by:

    1. Edward J. Bedrick, 2020. "Data reduction prior to inference: Are there consequences of comparing groups using a t‐test based on principal component scores?," Biometrics, The International Biometric Society, vol. 76(2), pages 508-517, June.
    2. Steland, Ansgar & von Sachs, Rainer, 2018. "Asymptotics for high-dimensional covariance matrices and quadratic forms with applications to the trace functional and shrinkage," Stochastic Processes and their Applications, Elsevier, vol. 128(8), pages 2816-2855.
    3. Neudecker, Heinz & Satorra, Albert, 1996. "The algebraic equality of two asymptotic tests for the hypothesis that a normal distribution has a specified correlation matrix," Statistics & Probability Letters, Elsevier, vol. 30(2), pages 99-103, October.
    4. 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.
    5. Shuangge Ma & Michael R. Kosorok & Jason P. Fine, 2006. "Additive Risk Models for Survival Data with High-Dimensional Covariates," Biometrics, The International Biometric Society, vol. 62(1), pages 202-210, March.
    6. J. O. Bauer & B. Drabant, 2021. "Regression based thresholds in principal loading analysis," Papers 2103.06691, arXiv.org, revised Mar 2022.
    7. Boik, Robert J., 2013. "Model-based principal components of correlation matrices," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 310-331.
    8. Haruhiko Ogasawara, 2002. "Concise formulas for the standard errors of component loading estimates," Psychometrika, Springer;The Psychometric Society, vol. 67(2), pages 289-297, June.
    9. Kollo, Tõnu & Ruul, Kaire, 2003. "Approximations to the distribution of the sample correlation matrix," Journal of Multivariate Analysis, Elsevier, vol. 85(2), pages 318-334, May.
    10. Liu, Shuangzhe & Leiva, Víctor & Zhuang, Dan & Ma, Tiefeng & Figueroa-Zúñiga, Jorge I., 2022. "Matrix differential calculus with applications in the multivariate linear model and its diagnostics," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    11. Chen, Dachuan, 2024. "High frequency principal component analysis based on correlation matrix that is robust to jumps, microstructure noise and asynchronous observation times," Journal of Econometrics, Elsevier, vol. 240(1).
    12. Bauer, Jan O. & Drabant, Bernhard, 2023. "Regression based thresholds in principal loading analysis," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    13. Haruhiko Ogasawara, 2003. "Oblique factors and components with independent clusters," Psychometrika, Springer;The Psychometric Society, vol. 68(2), pages 299-321, June.
    14. Bauer, Jan O. & Drabant, Bernhard, 2021. "Principal loading analysis," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    15. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2009. "Optimal rank-based testing for principal component," Working Papers ECARES 2009_013, ULB -- Universite Libre de Bruxelles.
    16. Steland, Ansgar & von Sachs, Rainer, 2016. "Asymptotics for High–Dimensional Covariance Matrices and Quadratic Forms with Applications to the Trace Functional and Shrinkage," LIDAM Discussion Papers ISBA 2016038, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    17. Choi, Jungjun & Yang, Xiye, 2022. "Asymptotic properties of correlation-based principal component analysis," Journal of Econometrics, Elsevier, vol. 229(1), pages 1-18.
    18. Aaron Fisher & Brian Caffo & Brian Schwartz & Vadim Zipunnikov, 2016. "Fast, Exact Bootstrap Principal Component Analysis for > 1 Million," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(514), pages 846-860, April.

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