Estimation and tests in finite mixture models of nonparametric densities
ESAIM: Probability and Statistics, Tome 13 (2009), pp. 276-282.

Dans les modèles de mélanges de densités non paramétriques, une question est de déterminer le comportement asymptotique d'estimateurs et de statistiques de test sur les composantes identifiables. Des modèles de mélanges non paramétriques d'un nombre connu de densités sont considérés. Des conditions pour l'identifiabilité et pour les convergences des paramètres et fonctions identifiables sont présentées. Le comportement des statistiques de test est décrit et des estimateurs des composantes des densités sont définis dans plusieurs cas.

The aim is to study the asymptotic behavior of estimators and tests for the components of identifiable finite mixture models of nonparametric densities with a known number of components. Conditions for identifiability of the mixture components and convergence of identifiable parameters are given. The consistency and weak convergence of the identifiable parameters and test statistics are presented for several models.

DOI : 10.1051/ps:2008014
Classification : 62G10, 62H10
Mots-clés : mixture models, nonparametric densities, test, weak convergence 
@article{PS_2009__13__276_0,
     author = {Pons, Odile},
     title = {Estimation and tests in finite mixture models of nonparametric densities},
     journal = {ESAIM: Probability and Statistics},
     pages = {276--282},
     publisher = {EDP-Sciences},
     volume = {13},
     year = {2009},
     doi = {10.1051/ps:2008014},
     mrnumber = {2528084},
     zbl = {1180.62049},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps:2008014/}
}
TY  - JOUR
AU  - Pons, Odile
TI  - Estimation and tests in finite mixture models of nonparametric densities
JO  - ESAIM: Probability and Statistics
PY  - 2009
SP  - 276
EP  - 282
VL  - 13
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps:2008014/
DO  - 10.1051/ps:2008014
LA  - en
ID  - PS_2009__13__276_0
ER  - 
%0 Journal Article
%A Pons, Odile
%T Estimation and tests in finite mixture models of nonparametric densities
%J ESAIM: Probability and Statistics
%D 2009
%P 276-282
%V 13
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps:2008014/
%R 10.1051/ps:2008014
%G en
%F PS_2009__13__276_0
Pons, Odile. Estimation and tests in finite mixture models of nonparametric densities. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 276-282. doi : 10.1051/ps:2008014. http://archive.numdam.org/articles/10.1051/ps:2008014/

[1] J.M. Azaïs, E. Gassiat and C. Mercadier, Asymptotic distribution and local power of the likelihood ratio test for mixtures: bounded and unbounded cases. Bernoulli 12 (2006) 775-799. | MR | Zbl

[2] K. Fukumizu, Likelihood ratio of unidendifiable models and multilayer neural networks, Ann. Statist. 31 (2003) 833-851. | MR | Zbl

[3] J.K. Ghosh and P.K. Sen, On the aymptotic performance of the log-likelihood ratio statistic for the mixture model and related results. In Proc. of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer, volume II, edited by L.M. Le Cam and R.A. Olshen, (1985), pp. 789-806. | MR

[4] M. Lemdani and O. Pons, Likelihood ratio tests in mixture models. C. R. Acad. Sci. Paris, Ser. I 322 (1995) 399-404. | MR | Zbl

[5] M. Lemdani and O. Pons, Likelihood ratio tests in contamination models, Bernoulli 5 (1999) 705-719. | MR | Zbl

[6] X. Liu and Y. Shao, Asymptotics for likelihood ratio tests under loss of identifiability, Ann. Statist. 31 (2003) 807-832. | MR | Zbl

[7] O. Pons, Estimation et tests dans les modèles de mélanges de lois et de ruptures. Hermès-Science Lavoisier, London and Paris (2009).

[8] A. Wald, Note on the consistency of the maximum likelihood estimate, Ann. Math. Statist. 20 (1949) 595-601. | MR | Zbl

Cité par Sources :