A METHOD WITH LOW COMPUTING COST TO APPROXIMATE INCREMENTS OF DETERMINANTS NEEDED FOR DERIVATIVE-FREE RESTRICTED MAXIMUM-LIKELIHOOD

Estimation of variance components by REML via derivative-free algorith ms requires the determinant of the coefficient matrix of mixed model e quations. A Monte Carlo-based method is proposed to approximate the in crement of the logarithm of the determinant of this coefficient matrix that corresponds to increments of the ratio between residual and addi tive-genetic variance. The computing cost of this method is linear wit h the order of the coefficient matrix. Results of approximate and exac t methods were compared. A bias, detected when the difference of the r atio between residual and additive genetic variance is large, vanishes as convergence is reached. The proposed procedure is accurate enough to estimate the maximum of the likelihood function, The REML estimates of variance components using the Monte Carlo approximation are also p resented.