 Solution

Let $P_k$ be orthogonal projection onto the column space of $X_k$, and let $\mu$ be the true mean. ThenUp to a common constant,The true model has $p_k=p^{\ast }$ and ${\lambda }_k=0$; every other candidate has $p_k>p^{\ast }$ and ${\lambda }_k\ge 0$. Its excess expected AIC is ${\lambda }_k+p_k-p^{\ast }>0$, while its excess expected BIC is ${\lambda }_k+(\log n-1)(p_k-p^{\ast })>0$ when $n>e$. Thus either minimum expected criterion selects the true model.