Minimax estimation of the Gaussian parametric regression

The paper considers the problem of estimating a $d\ge2$ dimensional mean vector of a multivariate normal distribution under quadratic loss. Let the observations be described by the equation
\begin{equation} Y=\theta+\sigma\xi, \end{equation}
where $\theta$ is a $d$-dimension vector of unknown parameters from some bounded set $\Theta\subset\mathbb R^d$, $\xi$ is a Gaussian random vector with zero mean and identity covariance matrix $I_d$, i.e. $Law(\xi)=\mathrm N_d(0,I_d)$ and $\sigma$ is a known positive number. The problem is to construct a minimax estimator of the vector $\theta$ from observations $Y$. As a measure of the accuracy of estimator $\hat\theta$ we select the quadratic risk defined as
$$ R(\theta,\hat\theta):=\boldsymbol E_\theta|\theta-\hat\theta|^2,\qquad|x|^2=\sum^d_{j=1}x^2_j, $$
where $\boldsymbol E_\theta$ is the expectation with respect to measure $\boldsymbol P_\theta$.
We propose a modification of the James–Stein procedure of the form
$$ \theta^*_+=\left(a-\frac c{|Y|}\right)_+Y, $$
where $c>0$ is a special constant and $a_+=\max(a,0)$ is a positive part of $a$. This estimate allows one to derive an explicit upper bound for the quadratic risk and has a significantly smaller risk than the usual maximum likelihood estimator and the estimator
$$ \theta^*=\left(1-\frac c{|Y|}\right)Y $$
for the dimensions $d\ge2$. We establish that the proposed procedure $\hat\theta_+$ is minimax estimator for the vector $\theta$.
A numerical comparison of the quadratic risks of the considered procedures is given. In conclusion it is shown that the proposed minimax estimator $\hat\theta_+$ is the best estimator in the mean square sense.