Minimax and shrinkage properties of the least-squares estimator

In the linear model (3.4.4), show that $\Sigma a_{i} \hat{\xi }_{i}$ (in the notation of Theorem 3.4.4) is minimax for estimating $\theta =\Sigma a_{i} \xi_{i}$ with squared error loss, under the restriction $\sigma^{2} \leq M$.

Consider the problem of estimating the mean based on $X \sim N_{r}(\boldsymbol {\theta }, I)$, where it is thought that $\theta_{i}=\sum_{j=1}^{s} \beta_{j} t_{i}^{j}$ where $(t_{i}, \ldots , t_{r})$ are known, $(\beta_{1}, \ldots , \beta_{s})$ are unknown, and $r-s>2$.

Consider the linear model with observed vector $X$ having distribution $N_n(Z\beta ,\sigma^2I_n)$, where $Z$ is an $n\times p$ known matrix, $p<n$, $\beta \in \mathbb {R}^{p}$, and $\sigma^2>0$, and the estimation of $l^\tau \beta$ under the squared error loss, where $l\in \mathcal R(Z)$. Show that the LSE $l^\tau \widehat{\beta }$ is minimax if $\sigma^2\in (0,c]$ with a constant $c$.