International Encyclopedia of Statistical Science

2011 Edition
| Editors: Miodrag Lovric

Skew-Symmetric Families of Distributions

  • Adelchi Azzalini
Reference work entry
DOI: https://doi.org/10.1007/978-3-642-04898-2_524

The term ‘skew-symmetric distributions’ refers to the construction of a continuous probability distribution obtained by applying a certain form of perturbation to a symmetric density function.

To be more specific, a concept of symmetric distribution must be adopted first, since in the multivariate setting various forms of symmetry have been introduced. The variant used in this context is the one of central symmetry, a natural extension of the traditional one-dimensional form to the d-dimensional case: if f0 is a density function on ℝd and ξ is a point of ℝd, central symmetry around ξ requires that f0(t − ξ) = f0( − t − ξ) for all t ∈ ℝd, ignoring sets of 0 probability. To avoid notational complications, we shall concentrate on the case with ξ = 0; it is immediate to rephrase what follows in the case of general ξ, which simply amounts to a shift of the location of the distribution.

If f 0 is a probability density function on ℝ dcentrally symmetric around 0, there are two largely...
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References and Further Reading

  1. Arellano-Valle RB, Branco MD, Genton MG (2006) A unified view on skewed distributions arising from selections. Canad J Stat 34:581–601MathSciNetMATHGoogle Scholar
  2. Azzalini A, Capitanio A (1999) Statistical applications of the multivariate skew normal distribution. J R Stat Soc B 61(3):579–602. Full version of the paper at http://arXiv.org (No. 0911.2093)
  3. Azzalini A, Capitanio A (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J R Stat Soc B 65(2):367–389. Full version of the paper at http://arXiv.org (No. 0911.2342)
  4. Branco MD, Dey DK (2001) A general class of multivariate skew-elliptical distributions. J Multivariate Anal 79(1):99–113MathSciNetMATHGoogle Scholar
  5. Genton MG (ed) (2004) Skew-elliptical distributions and their applications: a journey beyond normality. Chapman & Hall/CRC Press, Boca Raton, FLGoogle Scholar
  6. Genton MG, Loperfido N (2005) Generalized skew-elliptical distributions and their quadratic forms. Ann Inst Stat Math 57: 389–401MathSciNetGoogle Scholar
  7. Umbach D (2008) Some moment relationships for multivariate skew-symmetric distributions. Stat Probab Lett 78(12): 1619–1623MathSciNetMATHGoogle Scholar
  8. Wang J, Boyer J, Genton MG (2004) A skew-symmetric representation of multivariate distributions. Stat Sinica 14:1259–1270MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Adelchi Azzalini
    • 1
  1. 1.University of PaduaPaduaItaly