Journal of Mathematical Sciences

, Volume 167, Issue 4, pp 537–542 | Cite as

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.


Minimax Risk Quadratically Convex Linear Risk 


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Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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