RESTRICTED MAXIMUM-LIKELIHOOD-ESTIMATION OF COVARIANCES IN SPARSE LINEAR-MODELS

This paper discusses the restricted maximum likelihood (REML) approach for the estimation of covariance matrices in linear stochastic models , as implemented in the current version of the VCE package for covaria nce component estimation in large animal breeding models. The main fea tures are: 1) the representation of the equations in an augmented form that simplifies the implementation; 2) the parametrization of the cov ariance matrices by means of their Cholesky factors, thus automaticall y ensuring their positive definiteness; 3) explicit formulas for the g radients of the REML function for the case of large and sparse model e quations with a large number of unknown covariance components and poss ibly incomplete data, using the sparse inverse to obtain the gradients cheaply; 4) use of model equations that make separate formation of th e inverse of the numerator relationship matrix unnecessary. Many large scale breeding problems were solved with the new implementation, amon g them an example with more than 250 000 normal equations and 55 covar iance components, taking 41 h CPU time on a Hewlett Packard 755. (C) I nra/Elsevier, Paris.