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

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 ∈ Λε where Λε is a set of functions on R1 of the form Λε = {f : f ∈ ℱ, φ(f) ≥ Cψε}. Here ℱ 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, ψε is the minimax rate of testing and ε → 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.