Robust estimation on a parametric model via testing

TL;DR: In this article, the authors show that the estimator is robust even for models for which the maximum likelihood method is bound to fail, when the model is true and regular enough, at least when the number of observations is large.

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.