Define admissible, Bayes, minimax decision rules.

A random vector $X=\left(X_{1}, X_{2}, X_{3}\right)^{\mathrm{T}}$ has independent components, where $X_{i}$ has the normal distribution $\mathcal{N}\left(\theta_{i}, 1\right)$ when the parameter vector $\Theta$ takes the value $\theta=\left(\theta_{1}, \theta_{2}, \theta_{3}\right)^{\mathrm{T}}$. It is required to estimate $\Theta$ by a point $a \in \mathbb{R}^{3}$, with loss function $L(\theta, a)=\|a-\theta\|^{2}$. What is the risk function of the maximum-likelihood estimator $\widehat{\Theta}:=X ?$ Show that $\widehat{\Theta}$is dominated by the estimator $\widetilde{\Theta}:=\left(1-\|X\|^{-2}\right) X$.