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Spectral models for covariance matrices

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  • Robert J. Boik

Abstract

A new model for the simultaneous eigenstructure of multiple covariance matrices is proposed. The model is much more flexible than existing models and subsumes most of them as special cases. A Fisher scoring algorithm for computing maximum likelihood estimates of the parameters under normality is given. Asymptotic distributions of the estimators are derived under normality as well as under arbitrary distributions having finite fourth-order cumulants. Special attention is given to elliptically contoured distributions. Likelihood ratio tests are described and sufficient conditions are given under which the test statistics are asymptotically distributed as chi-squared random variables. Procedures are derived for evaluating Bartlett corrections under normality. Some conjectures made by Flury (1988) are verified; others are refuted. A small simulation study of the adequacy of the Bartlett correction is described and the new procedures are illustrated on two datasets. Copyright Biometrika Trust 2002, Oxford University Press.

Suggested Citation

  • Robert J. Boik, 2002. "Spectral models for covariance matrices," Biometrika, Biometrika Trust, vol. 89(1), pages 159-182, March.
    • Handle: RePEc:oup:biomet:v:89:y:2002:i:1:p:159-182
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    Cited by:

    1. Zongliang Hu & Zhishui Hu & Kai Dong & Tiejun Tong & Yuedong Wang, 2021. "A shrinkage approach to joint estimation of multiple covariance matrices," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(3), pages 339-374, April.
    2. Luca Bagnato & Antonio Punzo, 2021. "Unconstrained representation of orthogonal matrices with application to common principal components," Computational Statistics, Springer, vol. 36(2), pages 1177-1195, June.
    3. Pourahmadi, Mohsen & Daniels, Michael J. & Park, Trevor, 2007. "Simultaneous modelling of the Cholesky decomposition of several covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 98(3), pages 568-587, March.
    4. Alexander M. Franks, 2022. "Reducing subspace models for large‐scale covariance regression," Biometrics, The International Biometric Society, vol. 78(4), pages 1604-1613, December.
    5. Peter D. Hoff, 2009. "A hierarchical eigenmodel for pooled covariance estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(5), pages 971-992, November.
    6. Daniels, Michael J., 2006. "Bayesian modeling of several covariance matrices and some results on propriety of the posterior for linear regression with correlated and/or heterogeneous errors," Journal of Multivariate Analysis, Elsevier, vol. 97(5), pages 1185-1207, May.
    7. Nong Jin & Shiyu Zhou, 2006. "Data‐driven variation source identification for manufacturing process using the eigenspace comparison method," Naval Research Logistics (NRL), John Wiley & Sons, vol. 53(5), pages 383-396, August.
    8. Boik, Robert J., 2013. "Model-based principal components of correlation matrices," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 310-331.
    9. Bingkai Wang & Xi Luo & Yi Zhao & Brian Caffo, 2021. "Semiparametric partial common principal component analysis for covariance matrices," Biometrics, The International Biometric Society, vol. 77(4), pages 1175-1186, December.
    10. Pan, Yuqing & Mai, Qing, 2020. "Efficient computation for differential network analysis with applications to quadratic discriminant analysis," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    11. Juneja, Januj, 2012. "Common factors, principal components analysis, and the term structure of interest rates," International Review of Financial Analysis, Elsevier, vol. 24(C), pages 48-56.
    12. 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.
    13. Tsukuda, Koji & Matsuura, Shun, 2021. "Limit theorem associated with Wishart matrices with application to hypothesis testing for common principal components," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    14. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2011. "Optimal Rank-Based Tests for Common Principal Components," Working Papers ECARES ECARES 2011-032, ULB -- Universite Libre de Bruxelles.

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