Mean driven balance and uniformly best linear unbiased estimators

The equivalence of ordinary least squares estimators (OLSE) and Gauss–Markov estimators for models with variance–covariance matrix $$\sigma ^2{\mathbf M}$$ is extended to derive a necessary and sufficient balance condition for mixed models with mean vector $${\varvec{\mu }} = {{\mathbf X} {\varvec{\beta }}}$$ , with $${\mathbf {X}}$$ an incidence matrix, having OLSE for $$\varvec{\beta }$$ that are best linear unbiased estimator whatever the variance components. This approach leads to least squares like estimators for variance components. To illustrate the range of applications for the balance condition, interesting special models are considered.