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Journal of Mathematical Sciences

, Volume 152, Issue 6, pp 897–920 | Cite as

Estimation and detection of a function from tensor product spaces

  • Yu. I. Ingster
  • I. A. Suslina
Article
  • 46 Downloads

Abstract

We observe an unknown function of d variables ƒ(t), t ∈ [0, 1]d, in the white Gaussian noise of level ε > 0. We assume that {ie4526-01}, where {ie4526-02} is a ball in the Hilbert space {ie4526-03} of tensor product structure. Under minimax setup, we consider two problems: estimate ƒ (for quadratic losses) and detect ƒ, i.e., test the null hypothesis H0: ƒ = 0 against the alternatives {ie4526-04}. We are interested in the case {ie4526-05}. We study sharp, rate, and log-asymptotics (as ε → 0 and d → ∞) in the problems. In particular, we show that log-asymptotics are essentially different for d ≪ log ε−1 and d ≫ log ε−1. Bibliography: 19 titles.

Keywords

Russia Hilbert Space Null Hypothesis Information Technology Tensor Product 
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.

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Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.St.Petersburg State Electrotechnical UniversitySt.PetersburgRussia
  2. 2.St.Petersburg State University of Information Technologies, Mechanics, and OpticsSt.PetersburgRussia

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