We will develop new estimation methods in parametric models, using divergences and various representations of divergences. Our goal is to provide procedures with good robustness properties in contaminated samples and in the meantime characterized by high efficiency when the model is correctly specified. In this sense, we will construct adaptive procedures adequate for both contaminated and non-contaminated data. Since usually it is not known if the data set contains outliers or not, such adaptive procedures are very useful. They will be constructed using large families of divergences and selection rules based on efficiency criteria. Another way to achieve this goal is to define robustness and efficiency measures based on divergences and then to use these tools for defining optimal robust influence curves and corresponding estimators following the Hampel infinitesimal approach. Such technique will lead to equivariant optimal robust estimators. We will analyze the theoretical statistical properties of the new methods (consistency, asymptotic normality, robustness, efficiency) and we will compare these methods with other existing methods. We will also consider moment condition models for which we will study techniques for estimation and testing. New divergence based estimation methods will be developed for regression models and corresponding theoretical properties will be analyzed. Using such methods we will develop new robust methods in single index models for portfolio selection. Particularly, for these models, we will define and study new redescending M-estimators. In connection with regression models, robust and unbiased criteria for model selection will be studied.
All these theoretical elements together with corresponding computing methods will provide new tools useful for the statistical methodology, as well as for various economic applications, among them, portfolio optimization and estimation of financial risks.