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December 2018 Rho-estimators revisited: General theory and applications
Yannick Baraud, Lucien Birgé
Ann. Statist. 46(6B): 3767-3804 (December 2018). DOI: 10.1214/17-AOS1675

Abstract

Following Baraud, Birgé and Sart [Invent. Math. 207 (2017) 425–517], we pursue our attempt to design a robust universal estimator of the joint distribution of $n$ independent (but not necessarily i.i.d.) observations for an Hellinger-type loss. Given such observations with an unknown joint distribution $\mathbf{P}$ and a dominated model $\mathscr{Q}$ for $\mathbf{P}$, we build an estimator $\widehat{\mathbf{P}}$ based on $\mathscr{Q}$ (a $\rho$-estimator) and measure its risk by an Hellinger-type distance. When $\mathbf{P}$ does belong to the model, this risk is bounded by some quantity which relies on the local complexity of the model in a vicinity of $\mathbf{P}$. In most situations, this bound corresponds to the minimax risk over the model (up to a possible logarithmic factor). When $\mathbf{P}$ does not belong to the model, its risk involves an additional bias term proportional to the distance between $\mathbf{P}$ and $\mathscr{Q}$, whatever the true distribution $\mathbf{P}$. From this point of view, this new version of $\rho$-estimators improves upon the previous one described in Baraud, Birgé and Sart [Invent. Math. 207 (2017) 425–517] which required that $\mathbf{P}$ be absolutely continuous with respect to some known reference measure. Further additional improvements have been brought as 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. We also give some conditions for the maximum likelihood estimator to be a $\rho$-estimator. Finally, we consider the situation where the statistician has at her or his disposal many different models and we build a penalized version of the $\rho$-estimator for model selection and adaptation purposes. In the regression setting, this penalized estimator not only allows one to estimate the regression function but also the distribution of the errors.

Citation

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Yannick Baraud. Lucien Birgé. "Rho-estimators revisited: General theory and applications." Ann. Statist. 46 (6B) 3767 - 3804, December 2018. https://doi.org/10.1214/17-AOS1675

Information

Received: 1 June 2016; Revised: 1 November 2017; Published: December 2018
First available in Project Euclid: 11 September 2018

zbMATH: 1407.62169
MathSciNet: MR3852668
Digital Object Identifier: 10.1214/17-AOS1675

Subjects:
Primary: 62C20 , 62F99 , 62G05 , 62G07 , 62G08 , 62G35

Keywords: $\rho$-estimation , Density estimation , maximum likelihood estimators , metric dimension , regression with random design , robust estimation , statistical models , VC-classes

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 6B • December 2018
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