Model checks in inverse regression models with convolution-type operators

We consider the problem of testing parametric assumptions in an inverse regression model with a convolution-type operator. An L2-type goodness-of-fit test is proposed which com-pares the distance between a parametric and a nonparametric estimate of the regression function. Asymptotic normality of the corresponding test statistic is shown under the null hypothesis and under a general nonparametric alternative with different rates of convergence in both cases. The feasibility of the proposed test is demonstrated by means of a small sim-ulation study. In particular, the power of the test against certain types of alternative is investigated