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Consistent estimation of the order of mixture models. (English) Zbl 1081.62516

Summary: We consider the estimation of the number of components of mixture models using a maximum penalized likelihood method. B. G. Leroux [Ann. Stat. 20, No. 3, 1350–1360 (1992; Zbl 0763.62015)] proved, under some assumptions, that the method leads to an estimator which, asymptotically, does not underestimate the number of components a.s. We prove here the almost sure consistency of the maximum penalized likelihood estimator for an appropriate penalization sequence. The proof uses the locally conic parameterization developed by D. Dacunha-Castelle and É. Gassiat [ESAIM Probab. Stat. 1, 285–317 (1997; Zbl 1007.62507)]. A numerical study of the choice of the penalization term is proposed, as well as a comparison with a moment method.

MSC:

62F12 Asymptotic properties of parametric estimators
62F10 Point estimation
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