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

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

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Let replacement tolerance mean the greatest number of observations that can be replaced while the estimator remains bounded. For $n=2m$, a sample with $m$ zeros and $m$ copies of $a$ is within $m$ replacements of both the all-zero sample and its translate by $a$. If an equivariant estimator tolerated $m$ replacements, it would remain within bounded distance of both $T(0^n)$ and $T(0^n)+a$, which is impossible as $a\to \infty$. Thus at most $m-1$ replacements are tolerable.

For $n=2m+1$, a sample with $m$ zeros and $m+1$ copies of $a$ is obtained from the all-zero sample by $m+1$ replacements and from the all-$a$ sample by $m$ replacements. Translation equivariance again forces breakdown by $m+1$ replacements. In both cases the finite-sample replacement breakdown point is at most

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The sample median attains the equivariant upper bound:For the Huber estimator, fewer than half the observations cannot overpower the bounded scores of the uncontaminated majority. Evaluating the estimating equation below $x_{(1)}-k$ or above $x_{(n)}+k$ makes every uncontaminated score have the same sign. A contaminating majority can balance these scores arbitrarily far away, soA $\gamma$-trimmed mean remains bounded while at most $\lfloor \gamma n\rfloor$ arbitrary observations are removed by each tail trim; one more arbitrarily large replacement survives. Hence

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If $v_T=\mathbb{E}[operatornameIF(X;T,N({\theta }_0,1){)}^2]$, asymptotic normality gives an asymptotically level-$\alpha$ test that rejects whenFor the median, $v_T=\pi /2$. For Huber,For the trimmed mean, with $q={\Phi }^{-1}(1-\gamma )$,These tests have bounded influence functions, so a small contamination proportion has bounded first-order effect on their statistics, asymptotic levels, and powers. Their finite-contamination protection is quantified by the breakdown points in part (c).

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