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


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.


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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:

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