On the canonical form of scale mixtures of skew-normal distributions

TL;DR

This paper introduces a canonical form for scale mixtures of multivariate skew-normal distributions, highlighting its properties, invariance, and methods for transformation, with detailed analysis of the skew t-distribution and related statistical measures.

Contribution

It defines the canonical form for scale mixtures of skew-normal distributions and provides a method to transform any such distribution into this form, enhancing analysis and interpretation.

Findings

Canonical form corresponds to an affine invariant coordinate system.

Method for linear transformation to canonical form is presented.

Analysis of multivariate skewness and kurtosis indices for skew t-distributions.

Abstract

The canonical form of scale mixtures of multivariate skew-normal distribution is defined, emphasizing its role in summarizing some key properties of this class of distributions. It is also shown that the canonical form corresponds to an affine invariant co-ordinate system as defined in Tyler \emph{et} al. (2009), and a method for obtaining the linear transform that converts a scale mixture of multivariate skew-normal distribution into a canonical form is presented. Related results, where the particular case of the multivariate skew $t$ distribution is considered in greater detail, are the general expression of the Mardia indices of multivariate skewness and kurtosis and the reduction of dimensionality in calculating the mode.