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RHO-ESTIMATORS REVISITED: GENERAL THEORY AND APPLICATIONS

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-01314781
Contributor : Yannick Baraud <>
Submitted on : Wednesday, June 15, 2016 - 5:04:11 PM
Last modification on : Tuesday, May 26, 2020 - 6:50:52 PM

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  • HAL Id : hal-01314781, version 2
  • ARXIV : 1605.05051

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

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