Estimation of parametric and semiparametric mixture models using phi-divergences

TL;DR

This paper introduces new robust estimation methods for parametric and semiparametric mixture models using phi-divergences, including a proximal-point algorithm, and demonstrates their advantages through theoretical analysis and simulations.

Contribution

It develops a novel robust estimator based on the dual formula of phi-divergences and introduces a new structure for mixture models incorporating prior information about the unknown component.

Findings

The new estimator shows improved robustness in simulations.

The proximal-point algorithm converges reliably for divergence-based estimators.

Incorporating prior information enhances estimation accuracy.

Abstract

The study of mixture models constitutes a large domain of research in statistics. In the first part of this work, we present phi-divergences and the existing methods which produce robust estimators. We are more particularly interested in the so-called dual formula of phi-divergences. We build a new robust estimator based on this formula. We study its asymptotic properties and give a numerical comparison with existing methods on simulated data. We also introduce a proximal-point algorithm whose aim is to calculate divergence-based estimators. We give some of the convergence properties of this algorithm and illustrate them on theoretical and simulated examples. In the second part of this thesis, we build a new structure for two-component mixture models where one component is unknown. The new approach permits to incorporate a prior linear information about the unknown component such as moment-type and L-moments constraints. We study the asymptotic properties of the proposed estimators. Several experimental results on simulated data are illustrated showing the advantage of the novel approach and the gain from using the prior information in comparison to existing methods which do not incorporate any prior information except for a symmetry assumption over the unknown component.