Abstract. 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.Résumé. Nousétudions le test du rapport de vraisemblance (TRV) pour des hypothèses sur la mesure mélangeante dans un mélange en présenceéventuelle 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.Mathematics Subject Classification. 62F05, 62F12, 62H10, 62H30.
The estimation of the extremal dependence structure is spoiled by the impact
of the bias, which increases with the number of observations used for the
estimation. Already known in the univariate setting, the bias correction
procedure is studied in this paper under the multivariate framework. New
families of estimators of the stable tail dependence function are obtained.
They are asymptotically unbiased versions of the empirical estimator introduced
by Huang [Statistics of bivariate extremes (1992) Erasmus Univ.]. Since the new
estimators have a regular behavior with respect to the number of observations,
it is possible to deduce aggregated versions so that the choice of the
threshold is substantially simplified. An extensive simulation study is
provided as well as an application on real data.Comment: Published at http://dx.doi.org/10.1214/14-AOS1305 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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