Robust Estimation in Finite Mixture Models

TL;DR

This paper introduces a robust estimator for finite mixture models that adapts to model misspecification, provides non-asymptotic bounds, and includes a data-driven method for selecting the number of mixture components.

Contribution

It develops a robust estimation approach with theoretical guarantees and a model selection procedure for finite mixture models, addressing misspecification issues.

Findings

Non-asymptotic deviation bounds for the estimator.

Risk bounds for mixture parameters under correct model specification.

A data-driven selection rule achieving oracle inequalities.

Abstract

We observe a $n$-sample, the distribution of which is assumed to belong, or at least to be close enough, to a given mixture model. We propose an estimator of this distribution that belongs to our model and possesses some robustness properties with respect to a possible misspecification of it. We establish a non-asymptotic deviation bound for the Hellinger distance between the target distribution and its estimator when the model consists of a mixture of densities that belong to VC-subgraph classes. Under suitable assumptions and when the mixture model is well-specified, we derive risk bounds for the parameters of the mixture. Finally, we design a statistical procedure that allows us to select from the data the number of components as well as suitable models for each of the densities that are involved in the mixture. These models are chosen among a collection of candidate ones and we show that our selection rule combined with our estimation strategy result in an estimator which satisfies an oracle-type inequality.