For $\theta_\gamma=(\gamma,0,\ldots,0)^T$, every moved observation has residual
$$
\gamma\tau-(\tau,0,\ldots,0)\theta_\gamma=0.
$$
Every original observation has
$$
|y_i-x_i^T\theta_\gamma|
=|y_i-\gamma x_{i1}|
\leq M_y+\gamma\max_i|x_{i1}|.
$$
Selecting any $h$ losses and adding $\lambda\|\theta_\gamma\|_1=\lambda\gamma$ gives
$$
L_{Z_{\gamma,\tau}}(\theta_\gamma)
\leq h\rho\!\left(M_y+\gamma\max_i|x_{i1}|\right)+\lambda\gamma.
$$


Solution

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

 Solution

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