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

Only $h-1$ original observations remain in $Z_{\gamma ,\tau }$, so the $h$ selected order statistics must include at least one moved observation. If $\Vert \theta {\Vert }_2\le \gamma -1$, then $|{\theta }_1|\le \gamma -1$, and its residual obeysMonotonicity of the symmetric loss, together with nonnegativity of the other loss and penalty terms, gives