## 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

Zbl 1373.62141
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### References:

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