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

• Published in: Probability theory and related fields 2000-05, Vol.117 (1), p.17-48
• Main Authors: LEPSKI, O. V, TSYBAKOV, A. B
• Format: Article
• Language: English
• Subjects: Asymptotic properties; Exact sciences and technology; Fixed points (mathematics); Hypotheses; Hypothesis testing; Mathematics; Minimax technique; Nonparametric inference; Nonparametric statistics; Probability; Probability and statistics; Sciences and techniques of general use; Statistical tests; Statistics; White noise
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s004400050265

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.