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We consider a multivariate regression (growth curve) model of the form $Y=XBZ+\epsilon$, $E\epsilon =0$, $var(vec\epsilon )=W\otimes \Sigma$, where $W={\sum }_{i=1}^k{\theta }_i{V}_i$ and ${\theta }_i$’s are unknown scalar covariance components. In the case of replicated observations, we derive the explicit form of the locally best estimators of the covariance components under normality and asymptotic confidence ellipsoids for certain linear functions of the first order parameters $\left\{B_{ij}\right\}$ estimating simultaneously the first and the second order parameters.

The Extended Growth Curve Model (ECGM) is a multivariate linear model connecting different multivariate regression models in sample subgroups through common variance matrix. It has the form: $$Y=\sum _{i=1}^k{X}_i{B}_i{Z}_i^{\text{'}}+e,vec\left(e\right)\sim N_{n\times p}\left(0,\Sigma \otimes I_n\right).$$ Here, matrices $X_i$ contain subgroup division indicators, and $Z_i$ corresponding regressors. If $k=1$, we speak about (ordinary) Growth Curve Model. The model has already its age (it dates back to 1964), but it has many important applications. That is why it is still intensively studied. Many articles investigating...