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A Proximal Point Algorithm for Minimum Divergence Estimators with Application to Mixture Models

Laboratoire de Statistique Théorique et Appliquée, Université Pierre et Marie CURIE, 4 place Jussieu, 75005 Paris, France
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This paper is an extended version of our paper published in the 2nd Conference on Geometric Science of Information, Palaiseau, France, 28–30 October 2015.
Academic Editors: Frédéric Barbaresco and Frank Nielsen
Entropy 2016, 18(8), 277; https://doi.org/10.3390/e18080277
Received: 11 June 2016 / Revised: 20 July 2016 / Accepted: 21 July 2016 / Published: 27 July 2016
(This article belongs to the Special Issue Differential Geometrical Theory of Statistics)
Estimators derived from a divergence criterion such as φ - divergences are generally more robust than the maximum likelihood ones. We are interested in particular in the so-called minimum dual φ–divergence estimator (MDφDE), an estimator built using a dual representation of φ–divergences. We present in this paper an iterative proximal point algorithm that permits the calculation of such an estimator. The algorithm contains by construction the well-known Expectation Maximization (EM) algorithm. Our work is based on the paper of Tseng on the likelihood function. We provide some convergence properties by adapting the ideas of Tseng. We improve Tseng’s results by relaxing the identifiability condition on the proximal term, a condition which is not verified for most mixture models and is hard to be verified for “non mixture” ones. Convergence of the EM algorithm in a two-component Gaussian mixture is discussed in the spirit of our approach. Several experimental results on mixture models are provided to confirm the validity of the approach. View Full-Text
Keywords: ϕ–divergences; robust estimation; EM algorithm; proximal-point algorithms; mixture models ϕ–divergences; robust estimation; EM algorithm; proximal-point algorithms; mixture models
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MDPI and ACS Style

Al Mohamad, D.; Broniatowski, M. A Proximal Point Algorithm for Minimum Divergence Estimators with Application to Mixture Models. Entropy 2016, 18, 277. https://doi.org/10.3390/e18080277

AMA Style

Al Mohamad D, Broniatowski M. A Proximal Point Algorithm for Minimum Divergence Estimators with Application to Mixture Models. Entropy. 2016; 18(8):277. https://doi.org/10.3390/e18080277

Chicago/Turabian Style

Al Mohamad, Diaa, and Michel Broniatowski. 2016. "A Proximal Point Algorithm for Minimum Divergence Estimators with Application to Mixture Models" Entropy 18, no. 8: 277. https://doi.org/10.3390/e18080277

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