Robust estimation on a parametric model via testing

TL;DR

This paper introduces a robust parametric density estimator based on testing, which provides non-asymptotic risk bounds, is resilient to model misspecification, and closely matches the efficiency of maximum likelihood estimators in regular models.

Contribution

It proposes a new robust estimation method using testing procedures, with proven risk bounds and robustness properties, outperforming MLE in certain challenging scenarios.

Findings

Estimator is robust against model misspecification.

Risk bounds are established under mild assumptions.

Estimator closely matches MLE in regular models with large samples.

Abstract

We are interested in the problem of robust parametric estimation of a density from $n$ i.i.d. observations. By using a practice-oriented procedure based on robust tests, we build an estimator for which we establish non-asymptotic risk bounds with respect to the Hellinger distance under mild assumptions on the parametric model. We show that the estimator is robust even for models for which the maximum likelihood method is bound to fail. A numerical simulation illustrates its robustness properties. When the model is true and regular enough, we prove that the estimator is very close to the maximum likelihood one, at least when the number of observations $n$ is large. In particular, it inherits its efficiency. Simulations show that these two estimators are almost equal with large probability, even for small values of $n$ when the model is regular enough and contains the true density.