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**Rho-estimators revisited: general theory and applications.**
*(English)*
Zbl 1407.62169

Summary: Following Y. Baraud et al. [Invent. Math. 207, No. 2, 425–517 (2017; Zbl 1373.62141)], 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 [loc. cit.] 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.

### MSC:

62G35 | Nonparametric robustness |

62G05 | Nonparametric estimation |

62G07 | Density estimation |

62G08 | Nonparametric regression and quantile regression |

62C20 | Minimax procedures in statistical decision theory |

### Keywords:

\(\rho\)-estimation; robust estimation; density estimation; regression with random design; statistical models; maximum likelihood estimators; metric dimension; VC-classes### Citations:

Zbl 1373.62141
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\textit{Y. Baraud} and \textit{L. Birgé}, Ann. Stat. 46, No. 6B, 3767--3804 (2018; Zbl 1407.62169)

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