An alternative derivation method of mixed model equations from best linear unbiased prediction (BLUP) and restricted BLUP of breeding values not using maximum likelihood

Two mixed model equations (MME) for best linear unbiased prediction (BLUP) of breeding values and for restricted BLUP of breeding values were derived by maximum likelihood from the joint normal probability distribution of the observations and breeding values. As a result, MME is actually more general than maximum likelihood because we can prove that each set of solutions of MME are identical to BLUP and restricted BLUP of breeding values and then it does not depend on normality. In the present study, the author shows deriving directly each MME from BLUP and restricted BLUP equations for breeding values without assuming the joint normal distribution of the data and random effects. However, if we cannot assume the multivariate normal density distribution of the estimated aggregate breeding value and each breeding value for selected traits, the response to selection by restricted BLUP may deviate from the expected values.