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

Table of contents
- Solution d

Solution

Only

h - 1

original observations remain in

Z_{γ, τ}

, so the

h

selected order statistics must include at least one moved observation. If

∥ θ ∥_{2} \leq γ - 1

, then

∣ θ_{1} ∣ \leq γ - 1

, and its residual obeys

∣ γ τ - τ θ_{1} ∣ = τ ∣ γ - θ_{1} ∣ \geq τ .

Monotonicity of the symmetric loss, together with nonnegativity of the other loss and penalty terms, gives

L_{Z_{γ, τ}} (θ) \geq ρ (τ) .

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