= Unknown-variance risk estimation in a saturated Gaussian model
{title2=$2\operatorname{rank}(P)=n$}
For a <normal distribution> $N_n(\mu,\sigma^2I)$ 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.
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