Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 219 / 1 / e / Solution

Differentiating the log-likelihood gives the unique stationary pointIts Hessian is diagonal with entries $-1/V_1$, $-G/V_2$, and $-N_{\mathrm{SN}}/{\sigma }_{\mathrm{SN}}^2$, so it is the unique maximum. The estimators are unbiased and have varianceswhich equal their Cramér-Rao lower bounds because the normal location statistics are efficient.

Since $\theta =M_0^{\ast }+\Delta M-\mathcal{M}$, invariance of the maximum-likelihood estimator givesIt is unbiased and normal with variance