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× This section is under development!

Recall $\norm{X}^2 = \sum X_i^2$.

$X = (X_1,\ldots,X_p)'$, where each $X_i \sim \mathcal{N}(\theta_i,1).$ The goal is to get a good estimator $\widehat{\theta} = \widehat{\theta}(X)$ in terms of loss $L(\widehat{\theta}, \theta) = \norm{\widehat{\theta}-\theta}^2$ (Euclidean norm).

Since the loss is random, consider the risk function $R(\widehat{\theta}, \theta) = \E{L(\widehat{\theta},\theta)}$

$\widehat{\theta}$ strictly dominates $\widetilde{\theta}$ if $R(\widehat{\theta}, \theta) \le R(\tilde{\theta}, \theta), \forall \theta$ and strictly so for some $\theta$.

$\widehat{\theta}$ is admissible if $\widehat{\theta}$ is not strictly dominated by any other $\widetilde{\theta}$.

What about $\widehat{\theta}_0:=X$, which is MLE and UMVUE? It is indeed admissible, but only for $p=1,2$. Surprisingly, it is inadmissible for $p\ge 3.$ James&Stein (1961) showed that

$\boxed{\widehat{\theta}_{JS} := \left( 1-\frac{p-2}{\norm{X}^2} \right)X}$

strictly dominates $\widehat{\theta}_0:=X.$

Proof: Notice that $\norm{X-\theta}^2 \sim \chi^2_p$. Risk of James-Stein:
$R(\widehat{\theta}_{JS},\theta) = \E\left\{ \norm{X-\frac{(p-2)X}{\norm{X}^2}- \theta}^2 \right\}$
$= \E\left\{ \norm{X-\theta}^2 \right\} - 2 \E\left\{ \norm{(X-\theta)\frac{(p-2)X}{\norm{X}^2}} \right\} + \E\left\{ \norm{\frac{(p-2)X}{\norm{X}^2}}^2 \right\}$
$= p - 2(p-2) \sum_i^p \E\left\{ \frac{(X_i-\theta_i)X_i}{\norm{X}^2} \right\} + (p-2)^2 \E\left\{ \frac{1}{\norm{X}^2} \right\}$
$= p - 2(p-2)(p-2) \E\left\{ \frac{1}{\norm{X}^2} \right\} + (p-2)^2 \E\left\{ \frac{1}{\norm{X}^2} \right\}$
$= p - (p-2)^2 \E\left\{ \frac{1}{\norm{X}^2} \right\} < p,$
where the fourth equality holds since

Instead of shrinking to $0$, could also shrink to $\theta_0$:

$\boxed{\widehat{\theta}_{JS} = \theta_0 + \left(1-\frac{p-2}{\norm{X}^2}\right)(X-\theta_0)}$

which in turn is strictly dominated by positive-part James-Stein:

$\boxed{\widehat{\theta}_{JS+} = \theta_0 + \left(1-\frac{p-2}{\norm{X}^2}\right)_+(X-\theta_0)}$