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

Suppose for contradiction that $\Vert \hat{\theta }(Z_{\gamma ,\tau }){\Vert }_2\le M$ uniformly in $\gamma ,\tau$. Choose $\gamma >M+1$, so part d applies to every fitted value and gives $L(\hat{\theta })\ge \rho (\tau )$. For this fixed $\gamma$, part c gives the competing boundwhich is independent of $\tau$. Since $\rho (\tau )\to \infty$, choose $\tau$ so that the lower bound exceeds this upper bound, contradicting optimality. Thus $n-h+1$ replacements can make the estimate unbounded, andTogether with part b, the replacement breakdown point is exactly $(n-h)/n$.