In its simplest reading, the term “skew-normal” refers to a family of continuous probability distributions on the real line having density function of form
where ϕ (⋅) and Φ( ⋅) denote the { N}(0, 1) density and cumulative distribution function, respectively, and α is a real parameter which regulates the shape of the density. The fact that (1) integrates to 1 holds by a more general result, given by Azzalini (1985), where ϕ and Φ are replaced by analogous functions for any choice of two distributions symmetric around 0.
It is immediate that the choice α = 0 lends the { N}(0, 1) distribution, and that, if Z is a random variable with density (1), denoted Z ∼ { S}N(α), then − Z ∼ { S}N (− α). Figure 1a displays ϕ(z; α) for a few choices of α; only positive values of this parameter are considered, because of the property just stated.
Some examples of skew-normal density...
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References and Further Reading
Arnold BC, Beaver RJ (2000) Hidden truncation models. Sankhyā A 62(1):22–35
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Azzalini A (1986) Further results on a class of distributions which includes the normal ones. Statistica 46(2):199–208
Azzalini A (2005) The skew-normal distribution and related multivariate families (with discussion) Scand J Stat 32:159–188 (C/R 189–200)
Azzalini A, Capitanio A (1999) Statistical applications of the multivariate skew normal distribution. J R Stat SocB 61(3):579–602 Full version of the paper at http://arXiv.org (No. 0911.2093)
Azzalini A, Dalla Valle A (1996) The multivariate skew-normal distribution. Biometrika 83:715–726
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Azzalini, A. (2011). Skew-Normal Distribution. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_523
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