Consider a normally distributed random vector $X \in \mathbb{R}^{p}$ modelled as $X \sim N\left(\theta, I_{p}\right)$ where $\theta \in \mathbb{R}^{p}, I_{p}$ is the $p \times p$ identity matrix, and where $p \geqslant 3$. Define the Stein estimator $\hat{\theta}_{S T E I N}$ of $\theta$.

Prove that $\hat{\theta}_{S T E I N}$ dominates the estimator $\tilde{\theta}=X$ for the risk function induced by quadratic loss

$$\ell(a, \theta)=\sum_{i=1}^{p}\left(a_{i}-\theta_{i}\right)^{2}, \quad a \in \mathbb{R}^{p}$$

Show however that the worst case risks coincide, that is, show that

$$\sup _{\theta \in \mathbb{R}^{p}} E_{\theta} \ell(X, \theta)=\sup _{\theta \in \mathbb{R}^{p}} E_{\theta} \ell\left(\hat{\theta}_{S T E I N}, \theta\right)$$

[You may use Stein's lemma without proof, provided it is clearly stated.]