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.
Definition Let ϕ ( x ) {\displaystyle \phi (x)} denote the standard normal probability density function
ϕ ( x ) = 1 2 π e − x 2 2 {\displaystyle \phi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}}
with the cumulative distribution function given by
Φ ( x ) = ∫ − ∞ x ϕ ( t ) d t = 1 2 [ 1 + erf ( x 2 ) ] , {\displaystyle \Phi (x)=\int _{-\infty }^{x}\phi (t)\ \mathrm {d} t={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right],}
where "erf" is the error function. Then the probability density function (pdf) of the skew-normal distribution with parameter α {\displaystyle \alpha } is given by
f ( x ) = 2 ϕ ( x ) Φ ( α x ) . {\displaystyle f(x)=2\phi (x)\Phi (\alpha x).\,}
This distribution was first introduced by O'Hagan and Leonard (1976). Alternative forms to this distribution, with the corresponding quantile function, have been given by Ashour and Abdel-Hamid and by Mudholkar and Hutson. 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 ) {\displaystyle f(x)=2\phi (x)\Phi (x)} where ϕ ( ⋅ ) {\displaystyle \phi (\cdot )} is any PDF symmetric about zero and Φ ( ⋅ ) {\displaystyle \Phi (\cdot )} is any CDF whose PDF is symmetric about zero. To add location and scale parameters to this, one makes the usual transform x → x − ξ ω {\displaystyle x\rightarrow {\frac {x-\xi }{\omega }}} . One can verify that the normal distribution is recovered when α = 0 {\displaystyle \alpha =0} , and that the absolute value of the skewness increases as the absolute value of α {\displaystyle \alpha } increases. The distribution is right skewed if α > 0 {\displaystyle \alpha >0} and is left skewed if α < 0 {\displaystyle \alpha <0} . The probability density function with location ξ {\displaystyle \xi } , scale ω {\displaystyle \omega } , and parameter α {\displaystyle \alpha } becomes
f ( x ) = 2 ω ϕ ( x − ξ ω ) Φ ( α ( x − ξ ω ) ) . {\displaystyle f(x)={\frac {2}{\omega }}\phi {\left({\frac {x-\xi }{\omega }}\right)}\,\Phi {\left(\alpha \left({\frac {x-\xi }{\omega }}\right)\right)}.}
The skewness ( γ 1 {\displaystyle \gamma _{1}} ) of the distribution is limited to slightly less than the interval ( − 1 , 1 ) {\displaystyle (-1,1)} (see Estimation). As has been shown, the mode (maximum) m o {\displaystyle m_{o}} of the distribution is unique. For general α {\displaystyle \alpha } there is no analytic expression for m o {\displaystyle m_{o}} , but a quite accurate (numerical) approximation is:
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