Rho-estimators revisited: General theory and applications

TL;DR

This paper revisits rho-estimators, providing a general theory that enhances robustness, computational tractability, and applicability to regression and model selection, with improved risk bounds and handling of model misspecification.

Contribution

It introduces an improved, more general version of rho-estimators that work under weaker assumptions, with applications to regression, model selection, and aggregation.

Findings

Provides risk bounds close to minimax in many situations.

Extends rho-estimators to regression with random design.

Offers a computationally feasible aggregation procedure.

Abstract

Following Baraud, Birg\'e and Sart (2017), we pursue our attempt to design a robust universal estimator of the joint ditribution 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 $\mathscr{Q}$ 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, Birg\'e and Sart (2017) 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 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 to estimate the regression function but also the distribution of the errors.