{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:20:25Z","timestamp":1760242825900,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2016,7,27]],"date-time":"2016-07-27T00:00:00Z","timestamp":1469577600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Estimators derived from a divergence criterion such as     \u03c6 -    divergences are generally more robust than the maximum likelihood ones. We are interested in particular in the so-called minimum dual  \u03c6\u2013divergence estimator (MD\u03c6DE), an estimator built using a dual representation of \u03c6\u2013divergences. 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\u2019s 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 \u201cnon mixture\u201d 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.<\/jats:p>","DOI":"10.3390\/e18080277","type":"journal-article","created":{"date-parts":[[2016,7,27]],"date-time":"2016-07-27T09:38:46Z","timestamp":1469612326000},"page":"277","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Proximal Point Algorithm for Minimum Divergence Estimators with Application to Mixture Models"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8810-8571","authenticated-orcid":false,"given":"Diaa","family":"Al Mohamad","sequence":"first","affiliation":[{"name":"Laboratoire de Statistique Th\u00e9orique et Appliqu\u00e9e, Universit\u00e9 Pierre et Marie CURIE, 4 place Jussieu, 75005 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michel","family":"Broniatowski","sequence":"additional","affiliation":[{"name":"Laboratoire de Statistique Th\u00e9orique et Appliqu\u00e9e, Universit\u00e9 Pierre et Marie CURIE, 4 place Jussieu, 75005 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,7,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"McLachlan, G.J., and Krishnan, T. (2007). The EM Algorithm and Extensions, Wiley.","DOI":"10.1002\/9780470191613"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1287\/moor.1030.0073","article-title":"An Analysis of the EM Algorithm and Entropy-Like Proximal Point Methods","volume":"29","author":"Tseng","year":"2004","journal-title":"Math. Oper. Res."},{"key":"ref_3","unstructured":"Chr\u00e9tien, S., and Hero, A.O. Generalized Proximal Point Algorithms and Bundle Implementations. Available online: http:\/\/www.eecs.umich.edu\/techreports\/systems\/cspl\/cspl-316.pdf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1080\/01630568708816257","article-title":"How good are the proximal point algorithms?","volume":"9","author":"Goldstein","year":"1987","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_5","unstructured":"Chr\u00e9tien, S., and Hero, A.O. (1998, January 16\u201321). Acceleration of the EM algorithm via proximal point iterations. 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