[Paper Review] On the Canonical Form of Scale Mixtures of Skew-Normal Distributions

Antonella Capitanio|arXiv (Cornell University)|Jul 3, 2012

Statistical Distribution Estimation and Applications15 references19 citations

TL;DR

This paper introduces a canonical form for scale mixtures of multivariate skew-normal distributions, enabling affine-invariant coordinate representation and simplifying computation of multivariate skewness and kurtosis. The key contribution is a general expression for Mardia's indices and a method to reduce dimensionality in mode calculation, with explicit solutions for the skew-normal and skew-t distributions using a transformation to canonical form.

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.

Motivation & Objective

• To define a canonical form for scale mixtures of multivariate skew-normal distributions that simplifies representation of key distributional properties.
• To establish that this canonical form corresponds to an affine invariant coordinate system, enhancing its utility in multivariate data analysis.
• To derive general expressions for Mardia's indices of multivariate skewness and kurtosis applicable to the entire class of scale mixtures of skew-normal distributions.
• To provide a method for dimensionality reduction in mode computation for these distributions, particularly for the skew-t case.
• To formally prove unimodality and provide a closed-form expression for the unique mode of the multivariate skew-t distribution.

Proposed method

• Define the canonical form via an affine transformation that aligns the skewness direction with the first coordinate, reducing the shape parameter to a single non-zero component.
• Use the transformation $ Z^* = C^{-1} ilde{Z} $, where $ ilde{Z} $ is a standardized version of $ Z $, to achieve a canonical form with $ \alpha^* = (\alpha_*, 0, \dots, 0)^\top $.
• Derive the mode of the canonical form by setting the gradient of the density to zero, leading to a system of equations solvable via symmetry and monotonicity arguments.
• For the skew-t distribution, solve the mode equation involving the Student-t CDF $ T_1 $ and PDF $ t_1 $, with a transformed argument $ w(y) $, to find the unique mode at $ (y_0^*, 0, \dots, 0)^\top $.
• Generalize the mode expression to arbitrary scale mixtures by reducing the problem to a one-dimensional integral equation involving the mixing variable's density $ f_S(s) $.
• Establish affine invariance by showing the canonical form is invariant under linear transformations, consistent with Tyler et al. (2009).

Experimental results

Research questions

• RQ1 How can a canonical form be defined for scale mixtures of multivariate skew-normal distributions to simplify the analysis of shape and tail behavior?
• RQ2 Does the canonical form correspond to an affine invariant coordinate system, and if so, how can it be constructed?
• RQ3 What is the general expression for Mardia’s indices of multivariate skewness and kurtosis in this class of distributions?
• RQ4 Can the mode of the multivariate skew-t distribution be uniquely determined and expressed in closed form?
• RQ5 To what extent can dimensionality be reduced in mode computation for scale mixtures of skew-normal distributions?

Key findings

• The canonical form of a scale mixture of multivariate skew-normal distributions is defined such that all components except the first are symmetric, simplifying shape analysis.
• The canonical form corresponds to an affine invariant coordinate system, as defined in Tyler et al. (2009), enabling robust multivariate data analysis.
• A general closed-form expression for Mardia’s indices of multivariate skewness and kurtosis is derived for the entire class of scale mixtures of skew-normal distributions.
• The unique mode of the multivariate skew-t distribution is shown to be $ M_0 = \xi + \frac{y_0^*}{\delta_*} \omega \delta $, where $ y_0^* $ solves a specific equation involving the Student-t CDF and PDF.
• The mode of the canonical form is always of the form $ (y_0^*, 0, \dots, 0)^\top $, enabling dimensionality reduction in mode computation.
• The existence and uniqueness of the mode for the skew-t distribution is proven by showing the mode equation has a unique non-negative solution due to monotonicity of the involved functions.