Reference Work Entry

International Encyclopedia of Statistical Science

pp 1342-1344


Skew-Normal Distribution

  • Adelchi AzzaliniAffiliated withUniversity of Padua
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
$$\phi (z;\alpha ) = 2\:\phi (z)\;\Phi (\alpha z),\qquad (-\infty <z <\infty ),$$
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.
Skew-Normal Distribution. Figure 1

Some examples of skew-normal density functi ...

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