Under <homoskedasticity>, write
$$
C=\sigma_{\mathrm{int}}^2+\sigma_m^2+\sigma_C^2,
\qquad H=\sigma_{\mathrm{int}}^2+\sigma_m^2.
$$
Then $\widehat M_0=\bar q$ and $\widehat\theta=\bar q-\bar r$. Both are <unbiased estimators>, and their <covariance matrix> is
$$
\boxed{
\operatorname{Cov}\begin{pmatrix}\widehat M_0\\\widehat\theta\end{pmatrix}
=\begin{pmatrix}
C/K&C/K\\
C/K&C/K+H/N
\end{pmatrix}.}
$$
Indeed the <Fisher information> is
$$
I(M_0,\theta)=
\begin{pmatrix}K/C+N/H&-N/H\\-N/H&N/H\end{pmatrix},
$$
and its inverse is exactly the displayed covariance matrix. The estimators therefore attain the multivariate <Cramer-Rao bound> and are <efficient estimators>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 219 / 1 / c / Solution

{wiki}

An efficient estimator attains the relevant lower bound on estimator variance, such as the <Cramer-Rao bound> in a regular parametric model.

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