 Distribution of the Akaike information criterion in a normal linear model

Consider the correctly specified fixed-design normal linear model $Y=X\beta +\epsilon$, with $X$ an $n\times p$ full-column-rank design matrix, $n>p$, and $\epsilon \sim N_n(0,{\sigma }^2{I}_n)$ with ${\sigma }^2>0$. Both $\beta$ and ${\sigma }^2$ are unknown, so the Akaike information criterion counts $p+1$ fitted parameters. If $\mathrm{RSS}=\Vert Y-X\hat{\beta }{\Vert }^2$ is the residual sum of squares, thenThe orthogonal-projection decomposition of a random variable with a multivariate normal distribution gives $\mathrm{RSS}/{\sigma }^2\sim {\chi }_{n-p}^2$. Substituting into $\mathrm{AIC}=-2\ell (\hat{\beta },{\hat{\sigma }}^2)+2(p+1)$ yieldsHere ${\chi }_{n-p}^2$ denotes a random variable, not a quantile. The maximum-likelihood estimator of the error variance uses divisor $n$; the unbiased residual variance estimator with divisor $n-p$ cannot be substituted into the maximized likelihood. If $n=p$ and $X$ is invertible, the fit has zero residual sum of squares, and the Gaussian likelihood is unbounded as ${\sigma }^2$ decreases to zero. There is then no finite positive-variance maximum-likelihood estimator to which the displayed Akaike information criterion law applies.