Likelihood ratio inequalities with applications to various mixtures

We give two simple inequalities on likelihood ratios. A first application is the consistency of the maximum-penalized marginal-likelihood estimator of the number of populations in a mixture with Markov regime. The. second application is the derivation of the asymptotic power of the likelihood ratio test under loss of identifiability for contiguous alternatives. Finally, we propose self-normalized score tests that have exponentially decreasing level and asymptotic power 1. (C) 2002 Editions scientifiques et medicales Elsevier SAS.