Only $h-1$ original observations remain in $Z_{\gamma,\tau}$, so the $h$ selected order statistics must include at least one moved observation. If $\|\theta\|_2\leq\gamma-1$, then $|\theta_1|\leq\gamma-1$, and its residual obeys
$$
|\gamma\tau-\tau\theta_1|
=\tau|\gamma-\theta_1|\geq\tau.
$$
Monotonicity of the symmetric loss, together with nonnegativity of the other loss and penalty terms, gives
$$
L_{Z_{\gamma,\tau}}(\theta)\geq\rho(\tau).
$$


Solution

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

 Solution

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