Minimax risk for quadratically convex sets

We consider the problem of estimating the vector $\theta=(\theta_1,\theta_2,\dots)\in\Theta\subset l_2$ on the observations $y_i=\theta_i+\sigma_i\mathbf x_i$, $ i=1,2,\dots$, where $\mathbf x_i$ are i.i.d. $\mathcal N(0,1)$, the parametric set $\Theta$ is compact, orthosymmetric, convex and quadratically convex. We show that in that case the minimax risk is not very different from $\sup\mathfrak R_L(\Pi)$, where $\mathfrak R_L(\Pi)$ is the minimax linear risk in the same problem with the parametric set $\Pi$ and $\sup$ is taken over all the hyperrectangles $\Pi\subset\Theta$. Donoho, Liu, and McGibbon (1990) have obtained this result for the case of equal $\sigma_i$, $i=1,2,\dots$. Bibl. – 4 titles.