Quadratic risk ($R(\theta ,\hat{\theta })=E_{\theta }\Vert \hat{\theta }-\theta {\Vert }^2$)

The risk of an estimator under squared Euclidean error is its expected squared distance from the true parameter. Pointwise risk domination can be strict even when the supremum risk agrees, because the supremum can be approached at parameters tending to infinity.

The mean-vector prediction risk measures squared Euclidean norm error in estimating the deterministic response mean. For independent future noise with covariance matrix ${\sigma }^2{I}_n$, prediction of a new noisy response adds $n{\sigma }^2$ to this risk. A Gaussian orthogonal projection matrix $P$ of rank $k$ has risk $\Vert (I-P)\mu {\Vert }^2+k{\sigma }^2$.

For $Y\sim N_n(\mu ,{\sigma }^{2}I)$ and a fixed orthogonal projection matrix $P$ of rank $k$, the displayed expression is an unbiased estimator of the mean-vector prediction risk whenever ${\hat{\sigma }}^2$ is an unbiased estimator of ${\sigma }^2$. Independence of its two terms is unnecessary. Comparing fixed models yields the Mallows Cp penalty, but minimizing unbiased estimates does not preserve unbiasedness after selection.

For a normal distribution $N_n(\mu ,{\sigma }^{2}I)$ with unrestricted $\mu \in {\mathbb{R}}^n$ and unknown ${\sigma }^2$, an integrable data-only unbiased estimator of the mean-vector prediction risk of a fixed rank-$k$ orthogonal projection matrix exists exactly when $2k=n$. To prove necessity, randomize the mean by independent Gaussian variance $v$: conditioning adds $(n-k)v$ to the squared bias term, whereas the marginal variance identity would add $kv$. When $2k=n$, the residual sum of squares itself is unbiased.