Skew Multiple Scaled Mixtures of Normal Distributions with Flexible Tail Behavior and Their Application to Clustering

The family of multiple scaled mixtures of multivariate normal (MSMN) distributions has been shown to be a powerful tool for modeling data that allow different marginal amounts of tail weight. An extension of the MSMN distribution is proposed through the incorporation of a vector of shape parameters, resulting in the skew multiple scaled mixtures of multivariate normal (SMSMN) distributions. The family of SMSMN distributions can express a variety of shapes by controlling different degrees of tailedness and versatile skewness in each dimension. Some characterizations and probabilistic properties of the SMSMN distributions are studied and an extension to finite mixtures thereof is also discussed. Based on a sort of selection mechanism, a feasible ECME algorithm is designed to compute the maximum likelihood estimates of model parameters. Numerical experiments on simulated data and three real data examples demonstrate the efficacy and usefulness of the proposed methodology.