Discussion of "Model-based clustering and classification with non-normal mixture distributions" by S.X. Lee and G.J. McLachlan

It is a great pleasure to have the chance of reading and commenting on this very interesting paper that provides a unified view on non-gaussian mixture models. It is a very hot topic that has recently been receiving increasing attention in the literature. This paper is especially welcome as it offers to the reader an up-to-date review, with interesting stimuli for reflection and further insight. We would like to comment on the clustering side of the work. The many examples discussed in the paper show how the choice of the distributional shape for the mixture components can affect the clustering performances of the corresponding mixture model. The clustering results are assessed by comparison with a priori known information about group membership through the Adjusted Rand Index (ARI) or the misclassification rate. But in real applications the group structure is unknown and the researcher is faced with the need to derive the “best” clustering with no a priori information on group membership. In the literature on model-based clustering, this problem is often viewed as a model selection problem and likelihood based criteria, such as BIC or ICL, are usually suggested. We wonder if such a strategy can effectively lead to the recovery of the “true” group structure and we try to give an answer through a simple simulation study (which has been performed using the R packages EMMIX-skew and EMMIX-uskew described in the paper). In particular we focus on the selection of the shape of the mixture components by comparing restricted skew t finite mixtures with unrestricted ones (which in the examples of the paper seem to give the best results). Starting from the Australian Institute of Sports data (Section 5.2 in Lee and McLachlan’s paper), a two-component mixture of unrestricted skew t is fitted and the corresponding parameter estimates are used to simulate 500 datasets (with sample size equal to 202, i.e. the same size as the original dataset). For each unit of these data sets, the generating mixture component is recorded and assumed to be the “true” class. Afterwards, on each sample, a two component mixture model is fitted both with restricted and unrestricted skew t components, thus leading to two partitions of the units into two groups. The agreement between each clustering result and the corresponding “true” classification is evaluated through the ARI.