Graphical models for skew‐normal variates

Abstract

This paper explores the usefulness of the multivariate skew-normal distribution in the context of graphical models. A slight extension of the family recently discussed by Azzalini & Dalla Valle (1996) and Azzalini & Capitanio (1999) is described, the main motivation being the additional property of closure under conditioning. After considerations of the main probabilistic features, the focus of the paper is on the construction of conditional independence graphs for skew-normal variables. Necessary and sufficient conditions for conditional independence are stated, and the admissible structures of a graph under restriction on univariate marginal distribution are studied. Finally, parameter estimation is considered. It is shown how the factorization of the likelihood function according to a graph can be rearranged in order to obtain a parameter based factorization

Multivariate extended skew-t distributions and related families

“…The ESN distribution has been studied in more detail by Capitanio et al (2003), although without noticing that lighter tails than the normal distribution could be obtained for certain values of τ . We examine this interesting property in this paper.…”

Multivariate log‐skew‐elliptical distributions with applications to precipitation data

“…This unfortunate property seems to vanish in the case of the skew-t distribution. Azzalini and Capitanio (2003) noted that the behavior of the profile log-likelihood function of the skew-t distribution is more regular and demonstrate it numerically with several datasets. Although there is no rigorous proof of it, Azzalini and Genton (2008) presented a theoretical insight into why the Fisher information matrix is not singular at α = 0 in the case of the multivariate skew-t distribution.…”

“…Azzalini and Dalla Valle (1996) proposed a multivariate analog of the univariate skew-normal distribution. Branco and Dey (2001) and Azzalini and Capitanio (2003) introduced the univariate and multivariate skew-t distributions, which extend the respective skew-normal distributions by allowing to control the tails of the distribution with the additional degrees of freedom parameter. A more detailed description of these and other skewed models may be found in the book edited by Genton (2004) and in the review by Azzalini (2005).…”

“…We denote a multivariate skew-normal random vector with PDF (2) as X ∼ SN d (ξ, Ω, α). For more details and applications of the multivariate skew-normal distribution, see Azzalini and Dalla Valle (1996), Azzalini and Capitanio (1999), Capitanio et al (2003), and Azzalini (2005).…”

“…We denote a multivariate skew-t random vector with PDF (3) as X ∼ ST d (ξ, Ω, α, ν). For more details and applications of the multivariate skew-t distribution see, for example, Azzalini and Capitanio (2003) and Azzalini and Genton (2008), among others.…”