Minimax risk over quadratically convex sets

We consider the problem of estimating a vector θ = (θ₁, θ₂,…) ∈ Θ ⊂ l ₂ from observations y _i = θ _i + σ _i x _i , i = 1, 2,…, where the random values x _i are N(0, 1), independent, and identically distributed, the parametric set Θ is compact, orthosymmetric, convex, and quadratically convex. We show that in that case, the minimax risk is not very different from $\sup {\Re_L}\left( \Pi \right)$, where ${\Re_L}\left( \Pi \right)$ is the minimax linear risk in the same problem with parametric set Π, and sup is taken over all the hyperrectangles Π ⊂ Θ. Donoho, Liu, and McGibbon (1990) have obtained this result for the case of equal σ _i , i = 1, 2,…. Bibliography: 4 titles.