For model $k$, let $X_k$ contain its $p_k$ active columns. With known noise variance one,
$$
\ell_k(\beta)=-\frac n2\log(2\pi)-\frac12\|Y-X_k\beta\|^2,
$$
and full column rank gives
$$
\boxed{\widehat\beta_k=(X_k^TX_k)^{-1}X_k^TY.}
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 218 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-218/2/c/solution