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On the unification of families of skew-normal distributions. (English) Zbl 1117.62051
A new parametric family of multivariate distributions is considered which extends the notion of skew-normal (SN) distributions. It is called a “unified skew-normal” (SUN) family. A SUN-distribution can be obtained as the conditional distribution \((U_1\,|\, U_0+\gamma>0)\) where \(U_0\in \mathbb R^m\), \(U_1\in \mathbb R^d\), \((U_0,U_1)\) is a Gaussian vector, \(\gamma\in \mathbb R^m\) is a nonrandom vector. The SUN family encompasses such extensions to the basic SN family as the closed SN of G. González-Farías, A. Domínguez-Molina and A. K. Gupta [J. Stat. Plann. Inference 126, No. 2, 521–534 (2004; Zbl 1076.62052)], the hierarchical SN of B. Liseo and N. Loperfido [Stat. Probab. Lett. 61, No. 4, 395–401 (2003; Zbl 1101.62342)], and the fundamental SN of R. B. Arellano-Valle and M. G. Genton [J. Multivariate Anal. 96, No. 1, 93–116 (2005; Zbl 1073.62049)]. Formulas for moments of SUN are derived. Possible extensions to skew-elliptical families are discussed.

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60E05 Probability distributions: general theory
62H10 Multivariate distribution of statistics
Software:
sn
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