Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 5J / c / Solution

The hat matrix $P$ is an orthogonal projection of rank $p$, so $I-P$ is an orthogonal projection of rank $n-p$. Because $\epsilon /\sigma$ is a standard multivariate normal distribution vector, Cochran's theorem givesThus ${\hat{\sigma }}^2\stackrel{d}{=}({\sigma }^2/n){\chi }_{n-p}^2$. Substituting this into the preceding formula gives the Distribution of the Akaike information criterion in a normal linear model: