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


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.


Russia Hilbert Space Null Hypothesis Information Technology Tensor Product 
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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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