Unconstrained parametrizations for variance-covariance matrices
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The estimation of variance-covariance matrices through optimization of an objective function, such as a log-likelihood function, is usually a difficult numerical problem. Since the estimates should be positive semi-definite matrices, we must use constrained optimization, or employ a parametrization that enforces this condition. We describe here five different parametrizations for variance-covariance matrices that ensure positive definiteness, thus leaving the estimation problem unconstrained. We compare the parametrizations based on their computational efficiency and statistical interpretability. The results described here are particularly useful in maximum likelihood and restricted maximum likelihood estimation in linear and non-linear mixed-effects models, but are also applicable to other areas of statistics.
- Anderson, T. W., Olkin, I. and Underhill, L. G. (1987) Generation of random orthogonal matrices. SIAM Journal on Scientific and Statistical Computing, 8(4), 625–9.
- Bates, D. M. and Watts, D. G. (1988) Nonlinear Regression Analysis and Its Applications. Wiley, New York.
- Dennis, Jr., J. E. and Schnabel, R. B. (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, NJ.
- Jennrich, R. I. and Schluchter, M. D. (1986) Unbalanced repeated measures models with structural covariance matrices. Biometrics, 42(4), 805–20.
- Jupp, D. L. B. (1978) Approximation to data by splines with free knots. SIAM Journal of Numerical Analysis, 15(2), 328–43.
- Laird, N. M. and Ware, J. H. (1982) Random-effects models for longitudinal data. Biometrics, 38, 963–74.
- Leonard, T. and Hsu, J. S. J. (1993) Bayesian inference for a covariance matrix. Annals of Statistics, 21, 1–25.
- Lindstrom, M. J. and Bates, D. M. (1988) Newton-Raphson and EM algorithms for linear mixed-effects models for repeated-measures data. Journal of the American Statistical Association, 83, 1014–22.
- Lindstrom, M. J. and Bates, D. M. (1990) Nonlinear mixed effects models for repeated measures data. Biometrics, 46, 673–87.
- Pinheiro, J. C. (1994) Topics in Mixed Effects Models. PhD thesis, University of Wisconsin-Madison.
- Pinheiro, J. C. and Bates, D. M. (1995) Model building for nonlinear mixed-effects models. Technical Report 91, Department of Biostatistics, University of Wisconsin-Madison.
- Rao, C. R. (1973) Linear Statistical Inference and Its Applications, 2nd edn. Wiley, New York.
- Thisted, R. A. (1988) Elements of Statistical Computing. Chapman & Hall, London.
- Unconstrained parametrizations for variance-covariance matrices
Statistics and Computing
Volume 6, Issue 3 , pp 289-296
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- Kluwer Academic Publishers
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- Unconstrained estimation
- variance-covariance components estimation
- Cholesky factorization
- matrix logarithm
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