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A decision rule is admissible if no other rule has risk function no greater at every parameter and strictly smaller at one parameter. Parameter estimation is one decision problem; its admissible rules are admissible estimators.

For a decision rule $\delta$ and loss $L$, its risk function is $R(\theta ,\delta )={\mathbb{E}}_{\theta }L(\theta ,\delta (X))$. It is an admissible estimator if there is no rule with risk everywhere no larger and somewhere strictly smaller. A Bayes decision rule minimizes $\int R(\theta ,\delta )\pi (d\theta )$ for a specified prior $\pi$. A minimax decision rule minimizes ${\sup }_{\theta }R(\theta ,\delta )$ over all rules.

Here the maximum-likelihood estimator is ${\delta }_0(X)=X$, and its squared-error risk isWrite the proposed estimator as $\delta (X)=X+g(X)$ with $g(x)=-x/\Vert x{\Vert }^2$ for $x\ne 0$, and assign any value at zero, an event of probability zero. Expanding the squared loss givesStein's lemma for a unit-covariance normal vector says $\mathbb{E}[(X_i-{\theta }_i)g_i(X)]=\mathbb{E}[{\partial }_i{g}_i(X)]$ when the relevant derivatives and expectations are integrable. With $r=\Vert x\Vert$,The singularity causes no difficulty in three dimensions: $r^{-2}$ is locally integrable against the volume element $r^{2}drd\Omega$, and the normal tails control infinity. More formally apply Stein's lemma first to $g_{ϵ}(x)=-x/(r^2+ϵ)$, then let $ϵ\downarrow 0$; its squared norm and derivative magnitudes are bounded by constants times $r^{-2}$, so dominated convergence justifies the limit. ConsequentlyThe expectation is finite and strictly positive. Thus this James–Stein estimator strictly dominates the maximum likelihood estimator, which is inadmissible under the specified squared-error loss.

An admissible estimator is one for which there is no estimator $\delta$ satisfyingfor every $\theta$, with strict inequality for at least one $\theta$. For $p\ge 3$, the James–Stein estimatorhas riskIt strictly dominates $X$, and hence

If $a>1$, compare withIts bias is the negative of the original bias, so the squared-bias terms agree, whileIts risk is therefore strictly smaller for every $\theta$. Thus
${\hat{\theta }}_{a,b}$ is not an admissible estimator.