RHO-ESTIMATORS REVISITED: GENERAL THEORY AND APPLICATIONS

Yannick Baraud (1) , Lucien Birgé (2)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

Following Baraud, Birgé and Sart (2014), we pursue our attempt to design a universal and robust estimation method based on independent (but not necessarily i.i.d.) observations. Given such observations with an unknown joint distribution P and a dominated model for P, we build an estimator P based on and measure its risk by an Hellinger-type distance. When P does belong to the model, this risk is bounded by some new notion of dimension which relies on the local complexity of the model in a vicinity of P. In most situations this bound corresponds to the minimax risk over the model (up to a possible logarithmic factor). When P does not belong to the model, its risk involves an additional bias term proportional to the distance between P and , whatever the true distribution P. From this point of view, this new version of ρ-estimators improves upon the previous one described in Baraud, Birgé and Sart (2014) which required that P be absolutely continuous with respect to some known reference measure. Further additional improvements have been brought compared to the former construction. In particular, it provides a very general treatment of the regression framework with random design as well as a computationally tractable procedure for aggregating estimators. Finally, we consider the situation where the Statistician has at disposal many different models and we build a penalized version of the ρ-estimator for model selection and adaptation purposes. In the regression setting, this penalized estimator not only allows to estimate the regression function but also the distribution of the errors.

Mots clés: density estimation; Hellinger robustness; regression framework; rho-estimators

Domaines: Statistiques [math.ST]

Yannick Baraud, Lucien Birgé. RHO-ESTIMATORS REVISITED: GENERAL THEORY AND APPLICATIONS. 2016. ⟨hal-01314781v2⟩