 Solution

Here $H=hI$, $h=(1+\lambda {)}^{-1}$, and prediction risk isIt decreases then increases, with $h^{\ast }=\Vert \mu {\Vert }^2/(\Vert \mu {\Vert }^2+n{\sigma }^2)$, equivalently ${\lambda }^{\ast }=n{\sigma }^2/\Vert \mu {\Vert }^2$. Its minimum is $n{\sigma }^2+n{\sigma }^2\Vert \mu {\Vert }^2/(\Vert \mu {\Vert }^2+n{\sigma }^2)$; the endpoint limits are $2n{\sigma }^2$ and $n{\sigma }^2+\Vert \mu {\Vert }^2$.