Resistant techniques for nonparametric regression, generalized linear and additive models

The motivation for our research is issued from the importance of modeling in statistical work. In fact, the description of the relation between two or more qualitative or quantitative variables is one of the major concern of a statistician, but it is also of interest to applied researchers in economics, social and medical sciences, biology and so on. In this thesis, we develop parametric and nonparametric statistical tools for modern regression from the point of view of robust statistics. The work can essentially be summarized in three parts. In the first one, we address the problem of the automatic selection of the smoothing parameter in nonparametric regression and propose two resistant criteria which can, for example, be applied to M-type smoothing splines. The proposals are based on robust predictive error criteria. The second part of the thesis is devoted to robust estimation and testing in the parametric framework of generalized linear models. A class of M-estimators and a family of robust test statistics are developed. The estimators proposed are a generalization of the classical quasi-likelihood estimators and the test statistics are based on a robust deviance function, which makes them a generalization of the quasi-deviance tests. The statistical properties of both estimators and tests are derived. Finally, the last part of this work merges the techniques developed in parametric and nonparametric regression into the framework of generalized additive models. In particular, the techniques for univariate resistant smoothing are used as building blocks in the generalized additive model, and we transfer the family of test statistics developed for generalized linear models to the setting of generalized additive models. The possibility of implementation of these techniques has been our concern through all these three steps.