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Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point
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  • Published: May 2000

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

  • O.V. Lepski1 &
  • A.B. Tsybakov2 

Probability Theory and Related Fields volume 117, pages 17–48 (2000)Cite this article

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  • 60 Citations

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Abstract.

For the signal in Gaussian white noise model we consider the problem of testing the hypothesis H 0 : f≡ 0, (the signal f is zero) against the nonparametric alternative H 1 : f∈Λɛ where Λɛ is a set of functions on R 1 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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Authors and Affiliations

  1. Université Aix-Marseille 1, CMI, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France, , , , , , FR

    O.V. Lepski

  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

    A.B. Tsybakov

Authors
  1. O.V. Lepski
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  2. A.B. Tsybakov
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Received: 23 February 1998 / Revised version: 6 April 1999 / Published online: 30 March 2000

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Lepski, O., Tsybakov, A. Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point. Probab Theory Relat Fields 117, 17–48 (2000). https://doi.org/10.1007/s004400050265

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  • Issue Date: May 2000

  • DOI: https://doi.org/10.1007/s004400050265

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Keywords

  • White Noise
  • Hypothesis Testing
  • Gaussian White Noise
  • Noise Model
  • Sobolev Class
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