Unconstrained parametrizations for variance-covariance matrices

The estimation of variance-covariance matrices in situations that involve the optimization of an objective function (e.g. a log-likelihood function) is usually a difficult numerical problem, since the resulting estimates should be positive semi-definite matrices. We can either use constrained optimization, or employ a parameterization that enforces this condition. We describe here five different parameterizations for variance-covariance matrices that ensure positive definiteness, while leaving the estimation problem unconstrained. We compare the parameterizations based on their computational efficiency and statistical interpretability. The results described here are particularly useful in maximum likelihood and restricted maximum likelihood estimation in mixed effects models, but are also applicable to other areas of statistics.