Some methodological aspects of validation of models in nonparametric regression

In this paper we describe some general methods for constructing goodness of t tests in nonparametric regression models. Our main concern is the development of statisticial methodology for the assessment (validation) of speci c parametric modelsM as they arise in various elds of applications. The fundamental idea which underlies all these methods is the investigation of certain goodness of t statistics (which may depend on the particular problem and may driven by di erent criteria) under the assumption that a speci ed 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 modelM holds true" (and their associated p values) bear various methodological aws. Hence, our suggestion is, always to accompany such a test by an analysis of the type II error, which is in goodness of t problems often the more serious one. We will 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 speci c parametric form or homoscedasticity in nonparametric regression models. AMS Subject Classi cation: Primary 62G05, Secondary 62G10, 62G30, 62G07