In this paper, we address the problem of testing hypotheses using maximum likelihood statistics in non identifiable models. We derive the asymptotic distribution under very general assumptions.The key idea is a local reparameterization, depending on the underlying distribution, which is called locally conic. This method enlights how the general model induces the structure of the limiting distribution in terms of dimensionality of some derivative space. We present various applications of the theory. The main application is to mixture models. Under very general assumptions, we solve completely the problem of testing the size of the mixture using maximum likelihood statistics. We derive the asymptotic distribution of the maximum likelihood statistic ratio which takes an unexpected form.
