A REPARAMETERIZATION TO IMPROVE NUMERICAL OPTIMIZATION IN MULTIVARIATE REML (CO)VARIANCE COMPONENT ESTIMATION

Multivariate restricted maximum likelihood (REML) (co)variance compone nt estimation using numerical optimization on the basis of Downhill-Si mplex (DS) or quasi-Newton (QN) procedures suffers from the problem of undefined 'covariance matrices' as are produced by the optimizers. So far, this problem has been dealt with by assigning 'bad' function val ues. For this procedure to work, it is implied that the information th is 'bad' function Value conveys is sufficient to avoid going in the sa me direction in the following optimization step. To a limited degree D S can cope with this situation. On the other hand QN usually breaks do wn if this situation occurs too frequently. This contribution analyzes the problem and proposes a reparameterization of the covariance matri ces to solve it. As a result., faster converging QN optimizers can be used, as they no longer suffer from lack of robustness. Four real data sets were analyzed using a multivariate model estimating between 17 a nd 30 (co)variance components simultaneously. Optimizing on the Choles ky factor instead of on the (co)variance components themselves reduced the computing time by a factor of 2.5 to more than 250, when comparin g the robust modified DS optimizer operating on the original covarianc e matrices to a QN optimizer using reparameterized covariance matrices .