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Asymptotic expansions of the distributions of estimators in canonical correlation analysis under nonnormality

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  • Ogasawara, Haruhiko

Abstract

Asymptotic expansions of the distributions of typical estimators in canonical correlation analysis under nonnormality are obtained. The expansions include the Edgeworth expansions up to order O(1/n) for the parameter estimators standardized by the population standard errors, and the corresponding expansion by Hall's method with variable transformation. The expansions for the Studentized estimators are also given using the Cornish-Fisher expansion and Hall's method. The parameter estimators are dealt with in the context of estimation for the covariance structure in canonical correlation analysis. The distributions of the associated statistics (the structure of the canonical variables, the scaled log likelihood ratio and Rozeboom's between-set correlation) are also expanded. The robustness of the normal-theory asymptotic variances of the sample canonical correlations and associated statistics are shown when a latent variable model holds. Simulations are performed to see the accuracy of the asymptotic results in finite samples.

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  • 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.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:9:p:1726-1750
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    1. 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.
    2. James Steiger & Michael Browne, 1984. "The comparison of interdependent correlations between optimal linear composites," Psychometrika, Springer;The Psychometric Society, vol. 49(1), pages 11-24, March.
    3. William Rozeboom, 1965. "Linear correlations between sets of variables," Psychometrika, Springer;The Psychometric Society, vol. 30(1), pages 57-71, March.
    4. 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.
    5. Ogasawara, Haruhiko, 2006. "Asymptotic expansion of the sample correlation coefficient under nonnormality," Computational Statistics & Data Analysis, Elsevier, vol. 50(4), pages 891-910, February.
    6. Wegelin, Jacob A. & Packer, Asa & Richardson, Thomas S., 2006. "Latent models for cross-covariance," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 79-102, January.
    7. Fang, C. & Krishnaiah, P. R., 1982. "Asymptotic distributions of functions of the eigenvalues of some random matrices for nonnormal populations," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 39-63, March.
    8. 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.
    9. Anderson, T. W., 1999. "Asymptotic Theory for Canonical Correlation Analysis," Journal of Multivariate Analysis, Elsevier, vol. 70(1), pages 1-29, July.
    10. Ledyard Tucker, 1958. "An inter-battery method of factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 23(2), pages 111-136, June.
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    Cited by:

    1. Ogasawara, Haruhiko, 2009. "Asymptotic expansions in mean and covariance structure analysis," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 902-912, May.
    2. Denis Agniel & Tianxi Cai, 2017. "Analysis of multiple diverse phenotypes via semiparametric canonical correlation analysis," Biometrics, The International Biometric Society, vol. 73(4), pages 1254-1265, December.
    3. Ogasawara, Haruhiko, 2010. "Accurate distribution and its asymptotic expansion for the tetrachoric correlation coefficient," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 936-948, April.
    4. Ogasawara, Haruhiko, 2009. "On the estimators of model-based and maximal reliability," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1232-1244, July.
    5. Ogasawara, Haruhiko, 2016. "Bias correction of the Akaike information criterion in factor analysis," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 144-159.

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