Nonparametric consistency for maximum likelihood estimation and clustering based on mixtures of elliptically-symmetric distributions

While there is abundant work on methodology, algorithms, and applications, a smaller body of literature has investigated the relationship between the clusters found by a method and the underlying data-generating mechanism. Assuming that the observed data set is generated by independent and identical observations from a probability law P, consistency concerns the relationship between P and the outcome of a method for random samples of a size converging to infinity. In cluster analysis, the clustering itself and/or distributional parameters characterising the clustering may be of interest. Here we will derive consistency results for model-based clustering, i.e., clustering based on probability mixture models. More precisely, the results will concern maximum likelihood (ML) estimators (MLE) of finite mixtures of distributions from elliptically symmetrical distribution (ESD) families such as the Gaussian distribution. Finite mixture models (FMM) are convex combinations of probability distributions suitable to represent inhomogeneous populations.