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

Table of contents
- Solution e

Solution

For

Σ = I_{d}

,

tr (Σ) = d

and

∥Σ ∥_{2} = 1

. Taking

k

of order

max {n ϵ, 1}

in part d gives

∥ μ - μ_{0} ∥_{2} = O_{p} (\frac{d}{n} + ϵ) .

For Gaussian data under Huber contamination, the Tukey median has the sharper dependence

O_{p} (\frac{d}{n} + ϵ)

up to universal constants in its valid contamination range. Both have the same sampling term

d / n

, but the Tukey median is linear rather than square-root in the contamination fraction.

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