Some new algorithms for computing restricted maximum likelihood estimates of variance components

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Abstract
Let y represent an n × 1 observable random vector that follows the general mixed linear model , where β is a p×l vector of unknown parameterss i is a qi ×1 unobservable random vector whose distribution is (multivariate normal with mean vector 0 and variance-covariance matrix , and e is an n × 1 unobservable random vector whose distribution is . The problem considered is that of computing restricted maximum likelihood (REML) estimates of . In general, closed-form expressions for the REML estimates do not exist, in which case the estimate must be computed by an iterative numerical method. Before applying an iterative algorithm, it may be advisable to reparameterize, to “linearize” the likelihood equations, or to eliminate one or more of the likelihood equations (by absorption). “Linearized” versions of two common algorithms, the method of successive approximations (MSA) and the Newton-Raphson (NR) algorithm, are proposed. Numerical results suggest that these algorithms improve on the MSA and the NR algorithm and are superior to other widely used algorithms like the method of scoring and the EM algorithm.

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