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The likelihood ratio test for general mixture modelswithor without structural parameter

Published online by Cambridge University Press:  21 July 2009

Jean-Marc Azaïs ,
Élisabeth Gassiat  and
Cécile Mercadier
Jean-Marc Azaïs
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219, Université Paul Sabatier, 31062 Toulouse Cedex 9, France; azais@cict.fr
Élisabeth Gassiat
Affiliation:
Équipe Probabilités, Statistique et Modélisation, UMR CNRS 8628, Université Paris-Sud, Bâtiment 425, Université de Paris-Sud, 91405 Orsay Cedex, France.
Cécile Mercadier
Affiliation:
Université de Lyon, Université Lyon 1, CNRS UMR 5208 Institut Camille Jordan, Bâtiment du Doyen Jean Braconnier, 43, bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France.
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Abstract

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

Keywords

Likelihood ratio testmixture modelsnumber ofcomponentslocal powercontiguity

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Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 13 , July 2009 , pp. 301 - 327
Copyright
© EDP Sciences, SMAI, 2009

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