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