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Probability Theory and Related Fields

, Volume 117, Issue 1, pp 17–48 | Cite as

Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point

  • O.V. Lepski
  • A.B. Tsybakov

Abstract.

For the signal in Gaussian white noise model we consider the problem of testing the hypothesis H0 : f≡ 0, (the signal f is zero) against the nonparametric alternative H1 : f∈Λɛ where Λɛ is a set of functions on R1 of the form Λɛ = {f : f∈?, ϕ(f) ≥ Cψɛ}. Here ? is a Hölder or Sobolev class of functions, ϕ(f) is either the sup-norm of f or the value of f at a fixed point, C > 0 is a constant, ψɛ is the minimax rate of testing and ɛ→ 0 is the asymptotic parameter of the model. We find exact separation constants C* > 0 such that a test with the given summarized asymptotic errors of first and second type is possible for C > C* and is not possible for C < C*. We propose asymptotically minimax test statistics.

Keywords

White Noise Hypothesis Testing Gaussian White Noise Noise Model Sobolev Class 
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-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • O.V. Lepski
    • 1
  • A.B. Tsybakov
    • 2
  1. 1.Université Aix-Marseille 1, CMI, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, FranceFR
  2. 2.UMR 7599, Laboratoire de Probabilités et Modèles Aléatoires, Case 188, 4 place Jussieu, 75252 Paris Cedex 05, France. e-mail: tsybakov@ccr.jussieu.fr.FR

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