Consistency for constrained maximum likelihood estimation and clustering based on mixtures of elliptically-symmetric distributions under general data generating processes

The consistency of the maximum likelihood estimator for mixtures of elliptically-symmetric distributions for estimating its population version is shown, where the underlying distribution P is nonparametric and does not necessarily belong to the class of mixtures on which the estimator is based. In a situation where P is a mixture of well enough separated but nonparametric distributions it is shown that the components of the population version of the estimator correspond to the well separated components of P. This provides some theoretical justification for the use of such estimators for cluster analysis in case that P has well separated subpopulations even if these subpopulations differ from what the mixture model assumes.