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 L ₂-type goodness-of-fit test is proposed which compares the distance between a parametric and a non-parametric estimate of the regression function. Asymptotic normality of the corresponding test statistic is shown under the null hypothesis and under a general non-parametric alternative with different rates of convergence in both cases. The feasibility of the proposed test is demonstrated by means of a small simulation study. In particular, the power of the test against certain types of alternative is investigated. Finally, an empirical example is provided, in which the proposed methods are applied to the determination of the shape of the luminosity profile of the elliptical galaxy NGC 5017.