Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 5K / Solution

For a model with $k$ fitted parameters, the Akaike information criterion iswhere $\ell$ is the log-likelihood evaluated at the maximum-likelihood estimate.

In the normal linear model with known ${\sigma }^2$,The maximum-likelihood estimator is the ordinary least-squares estimator, with fitted mean $\hat{\mu }=HY$, whereSince only the $p$ components of $\beta$ are fitted,After multiplying by ${\sigma }^2$ and discarding the model-independent constant, this is exactly Mallows Cp:

To compare it with prediction error, write $Y=\mu +\epsilon$ and $Y^{\ast }=\mu +{\epsilon }^{\ast }$, where $\mu =X\beta$, the two errors are independent, and both have covariance ${\sigma }^{2}I$. Since $H\mu =\mu$, $H$ is an orthogonal projection of rank $p$, andThe cross term has zero expectation, soOn the other hand,Therefore the unbiased prediction-error identity for ordinary least squares gives

Solved by gpt-5.6-sol high.