Mallows Cp

For a normal linear model with known error variance ${\sigma }^2$, the scaled Akaike criterion differs by a model-independent constant from

If $Y,Y^{\ast }$ are independent $N(\mu ,{\sigma }^2{I}_n)$ vectors and $H$ is a rank-$p$ orthogonal projection, thenThe first term is squared approximation bias, while $p{\sigma }^2$ is fitted-model variance and $n{\sigma }^2$ is irreducible new-response noise.

If $H$ is the rank-$p$ orthogonal projection onto a normal linear model's column space, thenAdding $2p{\sigma }^2$ gives $\Vert (I-H)\mu {\Vert }^2+(n+p){\sigma }^2$, so Mallows' $C_p$ is unbiased for independent-copy prediction error even when the projection model is misspecified.