Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 223 / 1 / a / Solution

The gross-error sensitivity is ${\gamma }^{\ast }(T,F)={\sup }_x|\mathrm{IF}(x;T,F)|$. At $N(\theta ,1)$, the influence function of the sample median has magnitude $1/(2\phi (0))=\sqrt{\pi /2}$, soFor the Huber location estimator, ${\psi }_k(u)=\max (-k,\min (u,k))$ and $\mathbb{E}{\psi }_k^{\prime }(Z)=\mathbb{P}(|Z|\le k)=2\Phi (k)-1$, givingFor the symmetric normal law, the trimmed population mean is $\theta$. Writing $q_{\gamma }={\Phi }^{-1}(1-\gamma )$ in the supplied influence function gives