Scale and shape mixtures of multivariate skew-normal distributions

Abstract

We introduce a broad and flexible class of multivariate distributions obtained by both scale and shape mixtures of multivariate skew-normal distributions. We present the probabilistic properties of this family of distributions in detail and lay down the theoretical foundations for subsequent inference with this model. In particular, we study linear transformations, marginal distributions, selection representations, stochastic representations and hierarchical representations. We also describe an EMtype algorithm for maximum likelihood estimation of the parameters of the model and demonstrate its implementation on a wind dataset. Our family of multivariate distributions unifies and extends many existing models of the literature that can be seen as submodels of our proposal.

Comparison and Classification of Flexible Distributions for Multivariate Skew and Heavy-Tailed Data

“…Another extension is broadening the universe of families. For example, Arellano-Valle et al [64] proposed multiple scale-shape mixtures of multivariate skew-normal distributions. Another example is to investigate the (uni)modality of the distributions considered.…”

A Selective Overview of Skew-Elliptical and Related Distributions and of Their Applications

“…Arellano-Valle et al [53] and Arellano-Valle et al [54] have studies shape mixtures of multivariate SN variables, in a number of variant constructions. Proceeding further in this direction, Arellano-Valle et al [55] have examined the more general construction with random mixtures of both the scale and the slant parameter of SN variates. The SSMSN class so generated encompasses a number of existing formulations, forming a wide subset of those generated by Proposition 3; see their Proposition 1 which must be interpreted in the sense stated in its preceding paragraph.…”

“…Given the purely numerical nature of these exercises, they cannot be described as 'applied work', but they provide illustrative examples for a wide range of methods. Notable references which make use of the AIS data include [10,11,19,32,53,55,59,68], but this list could be far longer. There are seldom cross-paper comparisons of the various methods that are applied to the AIS data.…”

Integral stochastic ordering of the multivariate normal mean-variance and the skew-normal scale-shape mixture models

“…and it is denoted by X ∼ SN d (µ, Σ, α). Now, we consider the multivariate scale-shape mixtures of skew-normal (SSMSN) distributions proposed recently by Arellano-Valle et al [1].…”

“…In this regard, Arellano-Valle et al [1] introduced a broad and flexible class of multivariate distributions obtained by both scale and shape mixtures of multivariate skew-normal distributions. This family of multivariate distributions unifies and extends many existing models in the literature such as scale mixtures of skew-normal distributions [15] and shape mixtures of skew-normal distributions [2].…”