Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 223 / 2 / b

Table of contents
- Solution b

Solution

For fixed residual

r = y_{i} - x_{i}^{⊤} θ

, choosing

γ_{i} = 0

costs

r^{2}

, while choosing

γ_{i} = r

costs

ℓ

. No other nonzero choice improves on

γ_{i} = r

. Thus

in f_{γ_{i}} {(r - γ_{i})^{2} + ℓ 1_{{γ_{i} \neq = 0}}} = min (r^{2}, ℓ) = ρ_{L} (r)

with

L = ℓ

. Summing over observations proves the equivalence.

Solved by gpt-5.6-sol high.

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