Bivariate Nonparametric Regression Models: Simulations and Application.

In applied statistics regression is one of the most used tool in establishing the relationship between a response and an explicatory variable. When small or moderate variations of the covariate involve great consequences in the response the choice of an appropriate parametric model is a crucial starting point; in such situations nonparametric methods may turn up to be helpful. In this paper we introduce the Nadaraya-Watson local linear kernel estimator, for which some peculiar properties are given, as an alternative to the linear regression estimator . Goodness of fit of the regression model is generally conducted through formal (i.e. the Durbin-Watson test) and informal (i.e. graphical tools such as ``Q-Q plots'', ``Cox distances'', ...) approaches based on the analysis of the residuals from the regression itself. We propose a pseudo likelihood test, for which p-values can easily be computed, to check the appropriateness of the linear regression model versus the Nadaraya-Watson local linear kernel estimator. When the linear regression model is rejected, we suggest to associate to the Nadaraya-Watson local linear kernel estimate ``variability bands'' based on bootstrap residuals resampling. The same framework in building a regression model may be extended to examine the validity of parametric models in more complex forms, such as Generalized Linear Models, to compare regression surfaces and it also may be applied to functional data analysis. Some simulated examples are given and two case studies, in industrial and clinical fields, are discussed. All graphics and analysis are performed implementing specific routines in R computing environment and they are available from the authors.