The likelihood ratio test for general mixture models with or without structural parameter
ESAIM: Probability and Statistics, Tome 13 (2009), pp. 301-327.

Nous étudions le test du rapport de vraisemblance (TRV) pour des hypothèses sur la mesure mélangeante dans un mélange en présence éventuelle d'un paramètre structurel, et ce dans toutes les situations possibles. Le résultat principal donne la distribution asymptotique du TRV sous des hypothèses qui ne sont pas loin d'être nécessaires. Nous donnons une solution détaillée pour le test d'une simple distribution contre un mélange avec application aux lois gaussiennes, Poisson et binomiales, ainsi que pour le test du nombre de populations dans un mélange fini avec un paramètre structurel.

This paper deals with the likelihood ratio test (LRT) for testing hypotheses on the mixing measure in mixture models with or without structural parameter. The main result gives the asymptotic distribution of the LRT statistics under some conditions that are proved to be almost necessary. A detailed solution is given for two testing problems: the test of a single distribution against any mixture, with application to gaussian, Poisson and binomial distributions; the test of the number of populations in a finite mixture with or without structural parameter.

DOI : https://doi.org/10.1051/ps:2008010
Classification : 62F05,  62F12,  62H10,  62H30
Mots clés : likelihood ratio test, mixture models, number of components, local power, contiguity
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author = {Aza{\"\i}s, Jean-Marc and Gassiat, \'Elisabeth and Mercadier, C\'ecile},
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pages = {301--327},
publisher = {EDP-Sciences},
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url = {http://www.numdam.org/articles/10.1051/ps:2008010/}
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Azaïs, Jean-Marc; Gassiat, Élisabeth; Mercadier, Cécile. The likelihood ratio test for general mixture models with or without structural parameter. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 301-327. doi : 10.1051/ps:2008010. http://www.numdam.org/articles/10.1051/ps:2008010/

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