The paper concludes our investigations in looking for the locally best linear-quadratic estimators of mean value parameters and of the covariance matrix elements in a special structure of the linear model (2 variables case) where the dispersions of the observed quantities depend on the mean value parameters. Unfortunately there exists no linear-quadratic improvement of the linear estimator of mean value parameters in this model.

In the paper necessary and sufficient conditions for the existence and an explicit expression for the Bayes invariant quadratic unbiased estimate of the linear function of the variance components are presented for the mixed linear model $𝐭=𝐗\beta +ϵ$, $𝐄\left(𝐭\right)=𝐗\beta$, $\mathrm{𝐕𝐚𝐫}\left(𝐭\right)={\mathbf{0}}_{\mathbf{1}}{𝐔}_{\mathbf{1}}+{\mathbf{0}}_{\mathbf{2}}{𝐔}_{\mathbf{2}}+{\mathbf{0}}_{\mathbf{3}}{𝐔}_{\mathbf{3}}$, with three unknown variance components in the normal case. An application to some examples from the analysis of variance is given.

The method of determining Bayesian estimators for the special ratios of variance components called the intraclass correlation coefficients is presented. The exact posterior distribution for these ratios of variance components is obtained. The approximate posterior mean of this distribution is also derived. All computations are non-iterative and avoid numerical integration.