 Mean-vector prediction risk

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 $\Vert (I-P)\mu {\Vert }^2+k{\sigma }^2$.