By optimality and part a,
$$
\lambda\|\widehat\theta(Z')\|_1
\leq L_{Z'}(\widehat\theta(Z'))
\leq L_{Z'}(0)
\leq h\rho(M_y).
$$
Since the <Euclidean norm> is at most the $L^1$ norm,
$$
\|\widehat\theta(Z')\|_2\leq\frac{h\rho(M_y)}\lambda
$$
uniformly over all replacements of at most $n-h$ observations. The distance from the original finite estimate is therefore uniformly bounded, proving
$$
\epsilon^*(Z,\widehat\theta)\geq\frac{n-h}{n}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-223/2/b/solution