 Unbiased Gaussian projection risk estimate

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.