Rejoinder to the discussion of "Model-based clustering and classification with non-normal mixture distributions"

Non-normal mixture distributions have received increasing attention in recent years. Finite mixtures of multivariate skew-symmetric distributions, in particular, the skew normal and skew t-mixture models, are emerging as promising extensions to the traditional normal and t-mixture models. Most of these parametric families of skew distributions are closely related, and can be classified into four forms under a recently proposed scheme, namely, the restricted, unrestricted, extended, and generalised forms. In this paper, we consider some of these existing proposals of multivariate non-normal mixture models and illustrate their practical use in several real applications. We first discuss the characterizations along with a brief account of some distributions belonging to the above classification scheme, then references for software implementation of EM-type algorithms for the estimation of the model parameters are given. We then compare the relative performance of restricted and unrestricted skew mixture models in clustering, discriminant analysis, and density estimation on six real datasets from flow cytometry, finance, and image analysis. We also compare the performance of mixtures of skew normal and t-component distributions with other non-normal component distributions, including mixtures with multivariate normal-inverse-Gaussian distributions, shifted asymmetric Laplace distributions and generalized hyperbolic distributions.