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

Under homoskedasticity, writeThen ${\hat{M}}_0=\bar{q}$ and $\hat{\theta }=\bar{q}-\bar{r}$. Both are unbiased estimators, and their covariance matrix isIndeed the Fisher information isand its inverse is exactly the displayed covariance matrix. The estimators therefore attain the multivariate Cramer-Rao bound and are efficient estimators.