Finite sample properties of parametric MMD estimation: robustness to misspecification and dependence

TL;DR

This paper investigates the finite sample properties of parametric MMD estimation, demonstrating its robustness to dependence and outliers, and provides theoretical and numerical analysis of the stochastic gradient descent algorithm used.

Contribution

It offers a theoretical analysis of the robustness of MMD-based estimators to dependence and outliers, along with an improved bound for the associated stochastic gradient descent algorithm.

Findings

Estimator is robust to dependence and outliers

Theoretical bounds for stochastic gradient descent are improved

Numerical simulations support robustness claims

Abstract

Many works in statistics aim at designing a universal estimation procedure, that is, an estimator that would converge to the best approximation of the (unknown) data generating distribution in a model, without any assumption on this distribution. This question is of major interest, in particular because the universality property leads to the robustness of the estimator. In this paper, we tackle the problem of universal estimation using a minimum distance estimator presented in Briol et al. (2019) based on the Maximum Mean Discrepancy. We show that the estimator is robust to both dependence and to the presence of outliers in the dataset. Finally, we provide a theoretical study of the stochastic gradient descent algorithm used to compute the estimator, and we support our findings with numerical simulations. ** The proof of Proposition 4.4 in the published version contains a mistake. The mistake is fixed here (and the bound is actually improved by a factor 2). **