Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 218 / 2 / d / Solution

The solid curve is the test errorand the dashed curve is the training error $n^{-1}\Vert Y-{\hat{Y}}_{\lambda }{\Vert }_2^2$. As the horizontal coordinate tends to $+\infty$, $\lambda \to \infty$, so every penalized slope tends to zero and ${\hat{Y}}_{\lambda }\to \overline{Y}1$. The two limits are thereforerespectively.

As the horizontal coordinate tends to $-\infty$, $\lambda \downarrow 0$ and the fit approaches the ordinary least squares fit $H_{0}Y$. Hence the solid curve tends to $n^{-1}\Vert Y^{\ast }-H_{0}Y{\Vert }_2^2$, which the plot shows is approximately $0.4$.