Asymptotic distribution and power of the likelihood ratio test for mixtures:
bounded and unbounded case.
Elisabeth Gassiat, Jean-Marc Azais and Cecile Mercadier
we consider the log-likelihood ratio test (LRT) for testing the number of components
in a mixture of populations in a parametric family. We provide the asymptotic
distribution of the LRT statistic under the null hypothesis as well as under contiguous
alternatives when the parameter set is bounded. Moreover, for the
simple contamination model
we prove that,
under general assumptions,
the asymptotic power under contiguous hypotheses
may be arbitrarily close to the asymptotic level
when the set of parameters is large enough. In the particular problem of normal
distributions, we prove that, when the unknown mean is not a priori bounded, the
asymptotic power under contiguous hypotheses is equal to the asymptotic level.