Nonparametric Testing for Hammerstein Systems

We examine the problem of testing a parametric form of the nonlinearity of Hammerstein systems against a nonparametric alternative. This article proposes test statistics relying on nonparametric orthogonal series regression function estimates. An asymptotic theory of the proposed tests is established including the asymptotic normality under the null hypothesis that the parametric form of the system nonlinearity is correct. Moreover, it is shown that under nonparametric alternatives, the probability of detecting that the parametric model is incorrect tends to one. The rate of detecting local alternatives is also established.