The likelihood ratio test for general mixture models with or without structural parameter

Abstract

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.

An asymptotic test for Quantitative Trait Locus detection in presence of missing genotypes

“…(2.15) where the observation X j stands for Y j , X j (t 1 ), X j (t 2 ) and where D is the set of scores defined in [4], see also [15] and [3]. A similar result is true under H 0 with a set D 0 .…”

“…It can be proved that assumptions 1, 2 and 3 of [4] hold. So, we can apply Theorem 1 of [4] and we have…”

Testing for homogeneity in mixture models

“…Recently Koenker and Mizera (2012) have noted that the Kiefer-Wolfowitz estimator can be formulated as a convex optimization problem and solved very efficiently by interior point methods. Recent work by Liu and Shao (2003) and Azaïs, Gassiat, and Mercadier (2009) has clarified the limiting behavior of the LRT for general classes of alternatives, and taken together these developments offer a fresh opportunity to explore the LRT for inference on mixtures.…”

“…A closely related approach was recently taken by Azaïs, Gassiat, and Mercadier (2009), who derive the distribution of the likelihood ratio test for a single distribution against arbitrary mixtures under fairly general conditions. Example 2.4.…”

“…This question has received considerable prior attention, again Lindsay (1995) provides an authoritative overview of this literature. However, more recently work by Liu and Shao (2003) and Azaïs, Gassiat, and Mercadier (2009) have revealed new features of the asymptotic behavior of the likelihood ratio for mixture settings that enable one to derive asymptotic critical values for the LRT.…”