Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 223 / 2 / b

Table of contents
- Solution b

Solution

By optimality and part a,

λ ∥ θ (Z^{'}) ∥_{1} \leq L_{Z^{'}} (θ (Z^{'})) \leq L_{Z^{'}} (0) \leq h ρ (M_{y}) .

Since the Euclidean norm is at most the

L^{1}

∥ θ (Z^{'}) ∥_{2} \leq \frac{h ρ ( M _{y} )}{λ}

uniformly over all replacements of at most

n - h

observations. The distance from the original finite estimate is therefore uniformly bounded, proving

ϵ^{*} (Z, θ) \geq \frac{n - h}{n} .

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