Advertisement

Journal of Mathematical Sciences

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

Minimax risk over quadratically convex sets

  • S. Reshetov
Article

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.

Keywords

Minimax Risk Quadratically Convex Linear Risk 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. N. Chentsov, “Estimation of an unknown density of distribution on observation,” Dokl. Akad. Nauk SSSR, 147, 45–48 (1962).MathSciNetGoogle Scholar
  2. 2.
    M. S. Pinsker, “The optimal filtration of square-integrable signals on the Gaussian noise,” Probl. Pered. Inform., 16(2), 52–68 (1980).MathSciNetGoogle Scholar
  3. 3.
    I. A. Ibragimov and R. Z. Has’minskii, “Nonparametric estimation of the value of a linear functional in Gaussian white noise,” Teor. Veroyatn. Primen., 29, 1–32 (1984).Google Scholar
  4. 4.
    D. L. Donoho, R. C. Liu, and B. MacGibbon, “Minimax risk over hyperrecttangles and implications,” Ann. Statist., 18, 1416–1437 (1990).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations