The <gross-error sensitivity> is $\gamma^*(T,F)=\sup_x|\operatorname{IF}(x;T,F)|$. At $N(\theta,1)$, the <influence function of the sample median> has magnitude $1/(2\varphi(0))=\sqrt{\pi/2}$, so
$$
\gamma^*(\widehat\theta_{\mathrm{med}})=\sqrt{\frac\pi2}.
$$
For the <Huber location estimator>, $\psi_k(u)=\max(-k,\min(u,k))$ and $\mathbb E\psi_k'(Z)=\mathbb P(|Z|\leq k)=2\Phi(k)-1$, giving
$$
\gamma^*(\widehat\theta_{\mathrm{Hub},k})
=\frac{k}{2\Phi(k)-1}.
$$
For the symmetric normal law, the trimmed population mean is $\theta$. Writing $q_\gamma=\Phi^{-1}(1-\gamma)$ in the supplied influence function gives
$$
\gamma^*(\widehat\theta_{\mathrm{trim},\gamma})
=\frac{q_\gamma}{1-2\gamma}.
$$

Solved by gpt-5.6-sol high.

Solution

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

At a distribution with positive density $f$ at its median $m$, the median influence function is $\operatorname{sgn}(x-m)/(2f(m))$ away from $m$.


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 Incoming links: Influence function of the sample median