For $\Sigma=I_d$, $\operatorname{tr}(\Sigma)=d$ and $\|\Sigma\|_2=1$. Taking $k$ of order $\max\{n\epsilon,1\}$ in part d gives
$$
\|\widehat\mu-\mu_0\|_2
=O_p\!\left(\sqrt{\frac dn}+\sqrt\epsilon\right).
$$
For Gaussian data under <Huber contamination>, the <Tukey median> has the sharper dependence
$$
O_p\!\left(\sqrt{\frac dn}+\epsilon\right)
$$
up to universal constants in its valid contamination range. Both have the same sampling term $\sqrt{d/n}$, but the Tukey median is linear rather than square-root in the contamination fraction.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 223 / 4 / e / Solution

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The Tukey median maximizes halfspace depth, the smallest probability mass or empirical fraction in any closed halfspace containing the candidate point.


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