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A Local Parameterization of Orthogonal and Semi-Orthogonal Matrices with Applications

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  • Boik, Robert J.

Abstract

This article describes a local parameterization of orthogonal and semi-orthogonal matrices. The parameterization leads to a unified approach for obtaining the asymptotic joint distributions of estimators of singular-values and -vectors, and of eigen-values and -vectors. The singular- or eigen-values can have arbitrary multiplicities. The approach is illustrated on principal components analyzes, canonical correlation analysis, inter-battery factory analysis, and reduced-rank regression.

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  • 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.
    • Handle: RePEc:eee:jmvana:v:67:y:1998:i:2:p:244-276
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    1. 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.
    2. Seo, T. & Kanda, T. & Fujikoshi, Y., 1995. "The Effects of Nonnormality of Tests for Dimensionality in Canonical Correlation and MANOVA Models," Journal of Multivariate Analysis, Elsevier, vol. 52(2), pages 325-337, February.
    3. Chamberlain, Gary, 1982. "Multivariate regression models for panel data," Journal of Econometrics, Elsevier, vol. 18(1), pages 5-46, January.
    4. 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.
    5. 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.
    6. Fujikoshi, Yasunori, 1974. "The likelihood ratio tests for the dimensionality of regression coefficients," Journal of Multivariate Analysis, Elsevier, vol. 4(3), pages 327-340, September.
    7. Douglas Clarkson & Robert Jennrich, 1988. "Quartic rotation criteria and algorithms," Psychometrika, Springer;The Psychometric Society, vol. 53(2), pages 251-259, June.
    8. 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.
    9. Izenman, Alan Julian, 1975. "Reduced-rank regression for the multivariate linear model," Journal of Multivariate Analysis, Elsevier, vol. 5(2), pages 248-264, June.
    10. Magnus, J.R. & Neudecker, H., 1979. "The commutation matrix : Some properties and applications," Other publications TiSEM d0b1e779-7795-4676-ac98-1, Tilburg University, School of Economics and Management.
    11. 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. Robert Boik, 2008. "Newton Algorithms for Analytic Rotation: an Implicit Function Approach," Psychometrika, Springer;The Psychometric Society, vol. 73(2), pages 231-259, June.
    2. 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.
    3. 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.
    4. 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.
    5. Boik, Robert J., 2013. "Model-based principal components of correlation matrices," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 310-331.
    6. Robert Boik, 2008. "Accurate confidence intervals in regression analyses of non-normal data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(1), pages 61-83, March.
    7. 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.
    8. Ogasawara, Haruhiko, 2006. "Asymptotic expansion of the sample correlation coefficient under nonnormality," Computational Statistics & Data Analysis, Elsevier, vol. 50(4), pages 891-910, February.
    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.

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