Some methodological aspects of validation of models in nonparametric regression

In this paper we describe some general methods for constructing goodness of fit tests in nonparametric regression models. Our main concern is the development of statisticial methodology for the assessment (validation) of specific parametric models M as they arise in various fields of applications. The fundamental idea which underlies all these methods is the investigation of certain goodness of fit statistics (which may depend on the particular problem and may be driven by different criteria) under the assumption that a specified model (which has to be validated) holds true as well as under a broad range of scenaria, where this assumption is violated. This is motivated by the fact that outcomes of tests for the classical hypothesis: "The model M holds true" (and their associated p values) bear various methodological flaws. Hence, our suggestion is always to accompany such a test by an analysis of the type II error, which is in goodness of fit problems often the more serious one. We give a careful description of the methodological aspects, the required asymptotic theory, and illustrate the main principles in the problem of testing model assumptions such as a specific parametric form or homoscedasticity in nonparametric regression models.