Statistics and Computing

, Volume 6, Issue 3, pp 289-296

First online:

Unconstrained parametrizations for variance-covariance matrices

  • José C. PinheiroAffiliated withDepartment of Biostatistics, University of Wisconsin-Madison
  • , Douglas M. BatesAffiliated withDepartment of Statistics, University of Wisconsin-Madison

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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.


Unconstrained estimation variance-covariance components estimation Cholesky factorization matrix logarithm