Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 218 / 1 / e / Solution

For prediction at covariate value $u$, the intercept contributes variance ${\sigma }^2/n$ in every case because $x$ is centered. The one-copy ridge shrinkage is $a_1=s/(s+\lambda )$, while the duplicated-design total has $a_2=2s/(2s+\lambda )$. ThusandSince $a_2>a_1$, duplication reduces ridge bias and increases variance.

For constrained Lasso, let $Z=z/s\sim N({\beta }_0,{\sigma }^2/s)$ and $C_t(Z)=\max (-t,\min (Z,t))$. Both designs have the identical fitted total $C_t(Z)$, so both haveDuplicating the predictor has no effect on Lasso predictions, despite making the coefficient vector nonunique.