Abstract: We show in a general mixed model the best linear unbiased estimators (BLUE) of fixed effects, with unknown variance components substituted by the REML estimates, are jointly asymptotically normal with the REML estimates. We also prove that given sufficient information the empirical distributions of the best linear unbiased predictors (BLUP) of random effects, again with REML-estimated variance components, converge to the true distributions of the corresponding random effects. As a consequence, we obtain a consistent estimate of the asymptotic variance-covariance matrix of the REML estimates. The results require neither that the data is normally distributed nor that the model is hierarchical (nested).
Key words and phrases: Asymptotic normality, empirical BLUP and BLUE, empirical distributions, mixed models, REML.