Dans (2) est developpee une theorie qui. sous certaines conditions de rcgularilc! donne la loi asymptotique du rapport de vraisemblance pour tester le nomhrc de: populations dans un melange. Nous donnons deux resultats, Tout d'ubord, on exhibc un exemple simple au la statistique du rapport de vraisemblance converge vers +00 avec unc probabilite de 1/2. C'est II notre connaissance, le premier exemple connu nun trivial de test de vraisemblance non convergent pour un ensemble compact de parametres, Ensuite, nous donnons des conditions de rcgularilc suffisantcs permettant d'appliquer les resultats de (2). © Academie des Sciences/Elsevier. Paris
About the maximumlikelihood lest for mixtures In [2J a theory is developed which under some regularity conditions, Rives the asymplOtical distribution of the likelihood ratio for testing the number of populatkms in a mixture. We give here two results. First, we exhibit a simpleexample what' tlte
likelihood ratio converges to +00 with probability 1/2. Up to (llIr knowledge, it is ti,e first non-trivial example of a non-convergent likelihood test in a compact parameter space. Also, we give some enough regularity conditions allowingto ClPPI.V the results of [2J. © Academic des Sciences/Elsevier, Paris
Abridged English Version
Let :F = {fh', x) I'Y En. T compact, be a family of probability densities with respect to some positive measure von R and "Yo E I'. The set of all p-mixtures of densities of :F is the set {ip defined by: