Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution

By Adelchi Azzalini, A. Capitanio. 2003

1,542 citations recorded by the source corpus.

Summary. A fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non‐symmetric densities. The approach is sufficiently general to encompass some recent proposals in the literature, variously related to the skew normal distribution. The special case of skew elliptical densities is examined in detail, establishing connections with existing similar work. The final part of the paper specializes further to a form of multivariate skew t‐density. Likelihood inference for this distribution is examined, and it is illustrated with numerical examples.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a010c0a76724d9f4123b8e255ee2fb0b07c2c843

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In Statistics and Probability

Citations in the corpus

Robust Likelihood Methods Based on the Skew‐t and Related Distributions (2008, methodology): “Its density function at x ∈ Rd , in the form adopted by Azzalini & Capitanio (2003), is fST(x ; ξ, , α, ν) = 2 td (x − ξ ; , ν) T { α ω−1(x − ξ ) ( ν + d ν + Q(x) )1/2 ; ν + d } , (14) where Q(x) = (x − ξ ) −1(x − ξ ) and td (x ; , ν) = ((ν + d)/2)| |1/2(νπ )d/2 (ν/2) ( 1 + Q(x) ν )−(ν+d)/2 denotes…”
On the Ghoudi, Khoudraji, and Rivest test for extreme‐value dependence (2009, background): “The log-returns for gas, however, exhibit heteroscedasticity; they can be described by a GARCH(1, 6) model whose residuals Yt follow a skewed t-distribution (Azzalini & Capitanio, 2003).”
Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models? (2012, background): “Mean and variance of Y ~ ST (ξ, ω2, α) can be computed as follows (see Azzalini and Capitanio, 2003, also for higher moments): E Y , with 1 , (9) 22 2 )( YVar , where 1 1 2 1 2 .”
An alternative multivariate skew Laplace distribution: properties and estimation (2010, methodology): “The same dataset was considered by Lye and Martin (1993), Jones (2001), Azzalini and Capitanio (2003) , DiCiccio and Monti (2004), and Arslan and Genc (2008).”
Lambert W random variables—a new family of generalized skewed distributions with applications to risk estimation (2009, positive): “…in the skew-normal case [Azzalini and Capitanio (1999), Arellano-Valle and Azzalini (2006)], but the same concept has also been used for student-t [Azzalini and Capitanio (2003)] and Cauchy distributions [Arnold and Beaver (2000), Behboodian, Jamalizadeh and Balakrishnan (2006)], Received…”
Quantile Factor Models (2017, background): “Likewise, Ando and Bai (2020) (AB 2020, hereafter) report that the common factor structures explaining the asset return distributions in global financial markets since the subprime crisis are different in the lower and the upper tails.”
EMPIRICAL LIKELIHOOD FOR ESTIMATING EQUATIONS WITH MISSING VALUES (2009, methodology): “We first generate bivariate random vector (Xi,Ui) τ from a skewed bivariate t-distribution suggested in [2] with five degrees of freedom, mean (0,0) , shape parameter (4,1) and dispersion matrix ̄ = [ 1 0.”
Linear quantile mixed models (2013, background): “Distributions: N(0,ψ2), normal with mean 0 and variance ψ2; tg , t with g degrees of freedom; χ2 g , chi-square with g degrees of freedom; Stg , skew t with g degrees of freedom (Azzalini and Capitanio 2003)”
On a class of two-piece skew-normal distributions (2005, methodology): “Other models, introduced by Ma and Genton 8 and Azzalini and Capitanio 9 to accommodate more than one mode, are obtained by developing flexible class of skew-symmetric distributions.”
Skew scale mixtures of normal distributions: Properties and estimation (2011, methodology): “(4) Reasoning as Azzalini (1985) and Azzalini and Capitanio (2003), it is natural to construct univariate and multivariate distributions that combine skewness with heavy tails.”
Skewed distributions in finance and actuarial science: a review (2015, background): “It was introduced by Branco and Dey (2001), independently in the extended form by Adcock (2002), and, in a slightly different notation, by Azzalini and Capitanio (2003).”
Mean field variational bayes for elaborate distributions (2011, background): “Examples include Skew t (e.g., Azzalini and Capitanio 2003), Generalized Inverse Gaussian and Generalized Hyperbolic distributions.”
Bayesian Nonparametric Calibration and Combination of Predictive Distributions (2015, positive): “Assume that the data are generated from the dynamic mixture model (mAR) yt ∼ pSN (μ1 + φ1yt−1, σ1, ρ1) + (1 − p)SN (μ2 + φ2yt−1, σ2, ρ2) with ρi ∈ {−1, 1}, where SN (μ, σ, ρ) denotes a skew-normal (see Azzalini and Capitanio 2003) with μ, σ and ρ the location, scale and asymmetry parameters.”
Estimating copula densities through wavelets (2009, methodology): “The log-returns for gas, however, exhibit heteroscedasticity; they can be modeled by a GARCH(1 , 6) model whose residuals Y t follow a skewed t -distribution (Azzalini and Capitanio, 2003).”