On testing nonparametric hypotheses for analytic regression functions in Gaussian noise

We are interested in testing nonparametric hypotheses when the model of observation is given by the following stochastic differential equation: dYε(t) = f(t)dt+ εdW (t) where W is a double-sided standard Brownian motion and f an unknown function. It is required to distinguish the simple hypothesis H0: f = 0 against the composite alternative Hε: f ∈ Λε where Λε is a class of functions analytic in a strip of the complex plane around the real axis and satisfying some conditions. Λε is separated from zero by the value Cψ(ε) for some functionals. We consider two kinds of such functionals: the uniform norm over [0, 1] and the value of the function at a given point belonging to [0, 1]. Using the minimax approach, we find the minimax rate of testing hypotheses ψ(ε) and the exact value of constant C for the two problems. Key words: minimax signal detection, Gaussian white noise model, nonparametric al-ternative, analytical functions