Distribution of the Akaike information criterion in a normal linear model
= Distribution of the Akaike information criterion in a normal linear model
{c}
For a full-rank $n\times p$ <normal linear model> with error variance $\sigma^2$, the maximum-likelihood residual variance satisfies
$$
\widehat\sigma^2\overset d=\frac{\sigma^2}{n}\chi^2_{n-p}.
$$
Consequently
$$
\operatorname{AIC}\overset d=
n\log(\chi^2_{n-p})+n\bigl(\log(2\pi\sigma^2/n)+1\bigr)+2(p+1).
$$