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The coordinatewise median suffers a contamination error of order $ϵ\sqrt{d}$ with constant probability. In contrast, the Tukey median under an isotropic Gaussian model satisfies a high-probability Euclidean error bound of orderwith probability at least $1-\delta$, up to universal constants. Its contamination term is dimension-free. Thus coordinatewise estimation loses a factor $\sqrt{d}$ in its dependence on adversarial contamination.

For $\Sigma =I_d$, $\mathrm{tr}(\Sigma )=d$ and $\Vert \Sigma {\Vert }_2=1$. Taking $k$ of order $\max \{nϵ,1\}$ in part d givesFor Gaussian data under Huber contamination, the Tukey median has the sharper dependenceup 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.