= Stein risk identity for normal mean estimators
{c}
{title2=$R(\eta,W+g(W))=m+\mathbb E_\eta(\|g(W)\|^2+2\operatorname{div}g(W))$}
= Stein risk identity
{c}
{synonym}
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 $g(w)=-a w/\|w\|^2$ and $m>2$, the origin singularity is integrable and the inner boundary term is $O(\varepsilon^{m-2})$, so the same identity applies by excision. Its divergence is $-a(m-2)/\|w\|^2$, giving risk $m+[a^2-2a(m-2)]\mathbb E\|W\|^{-2}$.
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