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The James–Stein estimator in the specified unit-variance model iswith an arbitrary value at $X=0$, a probability-zero event. The risk of $X$ is $p$. Gaussian Stein identity states $E_{\theta }[(X_i-{\theta }_i)g_i(X)]=E_{\theta }[{\partial }_i{g}_i(X)]$ for suitably integrable differentiable functions; it extends to the present singular function by truncation, since $p>2$ makes $E\Vert X{\Vert }^{-2}$ finite.

Put $c=p-2$ and $g(x)=-cx/\Vert x{\Vert }^2$. Then $\Vert g{\Vert }^2=c^2/\Vert x{\Vert }^2$ and $\mathrm{div}g=-c(p-2)/\Vert x{\Vert }^2$. Expanding the quadratic risk and applying Stein's lemma yieldsThus domination is strict for every finite $\theta$.

To identify the worst-case risk, let $r=\Vert \theta \Vert \to \infty$. On $\Vert X\Vert >r/2$, the reciprocal square is at most $4/r^2$. On the complementary ball, the Gaussian density is at most $(2\pi {)}^{-p/2}{e}^{-r^2/8}$, so its reciprocal-square integral is bounded by a constant times $e^{-r^2/8}{r}^{p-2}$, tending to zero. Therefore $E_{\theta }\Vert X{\Vert }^{-2}\to 0$, and $R(\theta ,\hat{\theta })\to p$ from below. HenceThe second supremum is approached at infinity and need not be attained at a finite parameter.