 Solution

The risk function of an estimator $\hat{\theta }$ under the stated quadratic loss isThe multivariate normal density is proportional, as a function of $\theta$, toso maximum likelihood estimation gives ${\hat{\theta }}_{\mathrm{MLE}}=X$. Writing $X=\theta +Z$ with $Z\sim N_p(0,I_p)$,for every $\theta \in {\mathbb{R}}^p$.