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Second-order accurate inference on eigenvalues of covariance and correlation matrices

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  • Boik, Robert J.

Abstract

Edgeworth expansions and saddlepoint approximations for the distributions of estimators of certain eigenfunctions of covariance and correlation matrices are developed. These expansions depend on second-, third-, and fourth-order moments of the sample covariance matrix. Expressions for and estimators of these moments are obtained. The expansions and moment expressions are used to construct second-order accurate confidence intervals for the eigenfunctions. The expansions are illustrated and the results of a small simulation study that evaluates the finite-sample performance of the confidence intervals are reported.

Suggested Citation

  • Boik, Robert J., 2005. "Second-order accurate inference on eigenvalues of covariance and correlation matrices," Journal of Multivariate Analysis, Elsevier, vol. 96(1), pages 136-171, September.
  • Handle: RePEc:eee:jmvana:v:96:y:2005:i:1:p:136-171
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    References listed on IDEAS

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    1. Wang, Suojin, 1992. "General saddlepoint approximations in the bootstrap," Statistics & Probability Letters, Elsevier, vol. 13(1), pages 61-66, January.
    2. Bahjat F. Qaqish, 2003. "A family of multivariate binary distributions for simulating correlated binary variables with specified marginal means and correlations," Biometrika, Biometrika Trust, vol. 90(2), pages 455-463, June.
    3. Fujikoshi, Y., 1978. "Asymptotic expansions for the distributions of some functions of the latent roots of matrices in three situations," Journal of Multivariate Analysis, Elsevier, vol. 8(1), pages 63-72, March.
    4. Booth, J. G. & Hall, P. & Wood, A. T. A., 1994. "On the Validity of Edgeworth and Saddlepoint Approximations," Journal of Multivariate Analysis, Elsevier, vol. 51(1), pages 121-138, October.
    5. Robert J. Boik, 2003. "Principal component models for correlation matrices," Biometrika, Biometrika Trust, vol. 90(3), pages 679-701, September.
    6. Robert J. Boik, 2002. "Spectral models for covariance matrices," Biometrika, Biometrika Trust, vol. 89(1), pages 159-182, March.
    7. 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.
    8. Thomas J. DiCiccio, 2002. "Accurate confidence limits for scalar functions of vector M-estimands," Biometrika, Biometrika Trust, vol. 89(2), pages 437-450, June.
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    Cited by:

    1. Boik, Robert J., 2013. "Model-based principal components of correlation matrices," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 310-331.
    2. Boik, Robert J., 2008. "An implicit function approach to constrained optimization with applications to asymptotic expansions," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 465-489, March.

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