 Stein risk identity for normal mean estimators

For $W\sim N_m(\eta ,I)$, expand the squared estimation error and apply Gaussian integration by parts to each cross term. When $g$ satisfies the integrability and boundary conditions needed for that identity, this gives the displayed formula. For $g(w)=-aw/\Vert w{\Vert }^2$ and $m>2$, the origin singularity is integrable and the inner boundary term is $O({\epsilon }^{m-2})$, so the same identity applies by excision. Its divergence is $-a(m-2)/\Vert w{\Vert }^2$, giving risk $m+[a^2-2a(m-2)]\mathbb{E}\Vert W{\Vert }^{-2}$.