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Parameters ξ {displaystyle xi ,} location (real) ω {displaystyle omega ,} scale (positive, real) α {displaystyle alpha ,} shape (real) Support x ∈ ( − ∞ ; + ∞ ) {displaystyle xin (-infty ;+infty )!} PDF 1 ω π e − ( x − ξ ) 2 2 ω 2 ∫ − ∞ α ( x − ξ ω ) e − t 2 2 d t {displaystyle {rac {1}{omega pi }}e^{-{rac {(x-xi )^{2}}{2omega ^{2}}}}int _{-infty }^{alpha left({rac {x-xi }{omega }}ight)}e^{-{rac {t^{2}}{2}}} dt} CDF Φ ( x − ξ ω ) − 2 T ( x − ξ ω , α ) {displaystyle Phi left({rac {x-xi }{omega }}ight)-2Tleft({rac {x-xi }{omega }},alpha ight)} T ( h , a ) {displaystyle T(h,a)} is Owen's T function Mean ξ + ω δ 2 π {displaystyle xi +omega delta {sqrt {rac {2}{pi }}}} where δ = α 1 + α 2 {displaystyle delta ={rac {alpha }{sqrt {1+alpha ^{2}}}}} Variance ω 2 ( 1 − 2 δ 2 π ) {displaystyle omega ^{2}left(1-{rac {2delta ^{2}}{pi }}ight)} |
In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.
Contents
Definition
Let
with the cumulative distribution function given by
where erf is the error function. Then the probability density function (pdf) of the skew-normal distribution with parameter
This distribution was first introduced by O'Hagan and Leonard (1976). A popular alternative parameterization is due to Mudholkar and Hutson (2000), which has a form of the c.d.f. that is easily inverted such that there is a closed form solution to the quantile function.
A stochastic process that underpins the distribution was described by Andel, Netuka and Zvara (1984). Both the distribution and its stochastic process underpinnings were consequences of the symmetry argument developed in Chan and Tong (1986), which applies to multivariate cases beyond normality, e.g. skew multivariate t distribution and others. The distribution is a particular case of a general class of distributions with probability density functions of the form f(x)=2 φ(x) Φ(x) where φ() is any PDF symmetric about zero and Φ() is any CDF whose PDF is symmetric about zero.
To add location and scale parameters to this, one makes the usual transform
Note, however, that the skewness of the distribution is limited to the interval
Estimation
Maximum likelihood estimates for
where
The maximum (theoretical) skewness is obtained by setting
Concern has been expressed about the impact of skew normal methods on the reliability of inferences based upon them.
Differential equation
The differential equation leading to the pdf of the skew normal distribution is
with initial conditions