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Restricted Maximum Likelihood Estimation for Animal Models
Using Derivatives of the Likelihood}

K. Meyer and S.P. Smith

*Genetics, Selection, Evolution* **28** (1996) :
23-49.

### Abstract

Restricted Maximum Likelihood estimation using first and second
derivatives of the likelihood is described. It relies on the
calculation of derivatives without the need for large matrix inversion
using an automatic differentiation procedure. In essence, this is an
extension of the Cholesky factorisation of a matrix. A
reparameterisation is used to transform the constrained optimisation
problem imposed in estimating covariance components to an
unconstrained problem, thus making the use of Newton-Raphson and
related algorithms feasible. A numerical example is given to
illustrate calculations. Several modified Newton-Raphson and Method
of Scoring algorithms are compared for applications to analyses of
beef cattle data, and contrasted to a derivative-free algorithm.
**Keywords** : Restricted Maximum Likelihood, derivatives, algorithms,
variance component estimation

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K.Meyer, Nov. 30, 1996