Skew Dimension-Wise Scaled Mixtures of Normal Distributions and Their Applications in Model-Based Clustering

Abstract The main goal of this paper is to address scenarios in which the distribution of multivariate real-valued data exhibits skewness and diverse tail behavior across dimensions. The dimension-wise scaled mixtures of normal (DSMN) distributions have been shown to be effective in modeling data with varying degrees of tail heaviness by dimension. An extension of the DSMN distribution is introduced by incorporating a vector of shape parameters, leading to the skew dimension-wise scaled mixtures of normal (SDSMN) distributions. The SDSMN family offers flexibility in expressing a range of shapes by allowing control over tailedness and skewness in each dimension. This study examines the characteristics and probabilistic properties of SDSMN distributions, as well as explores their extension to finite mixtures thereof. An ECME algorithm is developed utilizing a selection mechanism to compute the maximum likelihood estimates of model parameters. Numerical experiments conducted on simulated data and four real datasets from various fields, including biometry and biomedicine, demonstrate the effectiveness and practicality of the proposed methodology.