Minimaxity of the usual multivariate normal mean estimator
= Minimaxity of the usual multivariate normal mean estimator
For $X\sim N_p(\theta,I_p)$ under <quadratic loss>, the estimator $\delta_0(X)=X$ has constant risk $p$ and is minimax. Gaussian priors $N_p(0,c^2I_p)$ have Bayes risks $pc^2/(1+c^2)$, which tend to $p$ and therefore give the matching minimax lower bound.