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Restricted maximum likelihood estimation of variance components: computational aspects

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Suppose that y is an n x 1 observable random vector, whose distribution is multivariate normal with mean vector X(alpha), where X is;a known n x p matrix of rank p* and (alpha) is a p x 1 vector of unknown parameters. The covariance matrix of y is taken to be;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);where Z(,1), ..., Z(,c) are known matrices and (gamma)(,1), ..., (gamma)(,c), (gamma)(,c+1) are unknown parameters. The parameter space for the vector (gamma) =;((gamma)(,1), ..., (gamma)(,c), (gamma)(,c+1))' is taken to be the set (OMEGA)(,1)* of (gamma)-values for which (gamma)(,c+1) > 0 and the matrix;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);is positive definite (i = 1, ..., c);The problem considered is that of computing a restricted maxi- mum likelihood (REML) estimate of (gamma), that is a point (gamma) at which the log-likelihood function L(,1)((gamma); y) associated with a set of n-p* linearly independent error contrasts attains its supremum over (OMEGA)(,1)*;Aside from certain special cases, (gamma) must be computed numeri- cally, generally by some iterative algorithm. Fourteen algorithms are devised, using the following six approaches: (1) Application of the Newton-Raphson method to the log-likelihood function or, equiva- lently, the application of Newton's root-finding method to the like- lihood equations; (2) Application of Newton's method to linearized likelihood equations, for example (when c = 1) to the equations (xi)(,i)((gamma)) (PAR-DIFF)L(,1)/(PAR-DIFF)(gamma)(,i) = 0 (i = 1, 2), where (xi)(,i)((gamma)) is a function of (gamma) chosen so that, in the special case of balanced data, (xi)(,i)((gamma)) (PAR-DIFF)L(,1)/(PAR-DIFF)(gamma)(,i) (i = 1, 2) are linear in (gamma); (3) Application of the Newton-Raphson method to a "concentrated" log-likelihood function; (4) The Method of Scor- ing; (5) The EM algorithm; and (6) The Method of Successive Approximations;Efficient procedures for computing the iterates of each algorithm are presented, and some techniques for restricting the iterates to the parameter space are discussed. Also, extensions to the case where;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);for known nonnegative-definite matrices A(,1), ..., A(,c) are devised;Numerical results on the performance of the fourteen iterative algorithms are given for each of four data sets. One very effective algorithm is that obtained by applying Newton's method to a set of linearized, concentrated likelihood equations.