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Cholesky Decompositions and Estimation of A Covariance Matrix: Orthogonality of Variance--Correlation Parameters

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  • Mohsen Pourahmadi

Abstract

Chen & Dunson ([3]) have proposed a modified Cholesky decomposition of the form σ = D L L′D for a covariance matrix where D is a diagonal matrix with entries proportional to the square roots of the diagonal entries of Σ and L is a unit lower-triangular matrix solely determining its correlation matrix. This total separation of variance and correlation is definitely a major advantage over the more traditional modified Cholesky decomposition of the form LD-super-2L′, (Pourahmadi, [13]). We show that, though the variance and correlation parameters of the former decomposition are separate, they are not asymptotically orthogonal and that the estimation of the new parameters could be more demanding computationally. We also provide statistical interpretation for the entries of L and D as certain moving average parameters and innovation variances and indicate how the existing likelihood procedures can be employed to estimate the new parameters. Copyright 2007, Oxford University Press.

Suggested Citation

  • Mohsen Pourahmadi, 2007. "Cholesky Decompositions and Estimation of A Covariance Matrix: Orthogonality of Variance--Correlation Parameters," Biometrika, Biometrika Trust, vol. 94(4), pages 1006-1013.
    • Handle: RePEc:oup:biomet:v:94:y:2007:i:4:p:1006-1013
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    Cited by:

    1. Cao, Xuan & Khare, Kshitij & Ghosh, Malay, 2020. "Consistent Bayesian sparsity selection for high-dimensional Gaussian DAG models with multiplicative and beta-mixture priors," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    2. Guney, Yesim & Arslan, Olcay & Yavuz, Fulya Gokalp, 2022. "Robust estimation in multivariate heteroscedastic regression models with autoregressive covariance structures using EM algorithm," Journal of Multivariate Analysis, Elsevier, vol. 191(C).
    3. Daniels, M.J. & Pourahmadi, M., 2009. "Modeling covariance matrices via partial autocorrelations," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2352-2363, November.
    4. Lam, Clifford & Feng, Phoenix & Hu, Charlie, 2017. "Nonlinear shrinkage estimation of large integrated covariance matrices," LSE Research Online Documents on Economics 69812, London School of Economics and Political Science, LSE Library.
    5. Yu, Jing & Nummi, Tapio & Pan, Jianxin, 2022. "Mixture regression for longitudinal data based on joint mean–covariance model," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    6. Dipankar Bose & A. K. Chatterjee & Samir Barman, 2016. "Towards dominant flexibility configurations in strategic capacity planning under demand uncertainty," OPSEARCH, Springer;Operational Research Society of India, vol. 53(3), pages 604-619, September.
    7. Clifford Lam & Phoenix Feng & Charlie Hu, 2017. "Nonlinear shrinkage estimation of large integrated covariance matrices," Biometrika, Biometrika Trust, vol. 104(2), pages 481-488.
    8. Feng, Sanying & Lian, Heng & Xue, Liugen, 2016. "A new nested Cholesky decomposition and estimation for the covariance matrix of bivariate longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 102(C), pages 98-109.
    9. Lam, Clifford, 2020. "High-dimensional covariance matrix estimation," LSE Research Online Documents on Economics 101667, London School of Economics and Political Science, LSE Library.
    10. Lee, Keunbaik & Baek, Changryong & Daniels, Michael J., 2017. "ARMA Cholesky factor models for the covariance matrix of linear models," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 267-280.
    11. Xu, Lin & Xiang, Sijia & Yao, Weixin, 2019. "Robust maximum Lq-likelihood estimation of joint mean–covariance models for longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 397-411.
    12. Ying C. MacNab, 2018. "Some recent work on multivariate Gaussian Markov random fields," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(3), pages 497-541, September.
    13. Mohammadi, Raziyeh & Kazemi, Iraj, 2022. "A robust linear mixed-effects model for longitudinal data using an innovative multivariate skew-Huber distribution," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    14. Zhou Lan & Brian J. Reich & Joseph Guinness & Dipankar Bandyopadhyay & Liangsuo Ma & F. Gerard Moeller, 2022. "Geostatistical modeling of positive‐definite matrices: An application to diffusion tensor imaging," Biometrics, The International Biometric Society, vol. 78(2), pages 548-559, June.
    15. Fisher, Thomas J. & Sun, Xiaoqian, 2011. "Improved Stein-type shrinkage estimators for the high-dimensional multivariate normal covariance matrix," Computational Statistics & Data Analysis, Elsevier, vol. 55(5), pages 1909-1918, May.
    16. Weiping Zhang & Feiyue Xie & Jiaxin Tan, 2020. "A robust joint modeling approach for longitudinal data with informative dropouts," Computational Statistics, Springer, vol. 35(4), pages 1759-1783, December.
    17. Lu, Fei & Xue, Liugen & Cai, Xiong, 2020. "GEE analysis in joint mean-covariance model for longitudinal data," Statistics & Probability Letters, Elsevier, vol. 160(C).
    18. Yixin Chen & Weixin Yao, 2017. "Unified Inference for Sparse and Dense Longitudinal Data in Time-varying Coefficient Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(1), pages 268-284, March.

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