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{{Probability distribution |
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  name      =Skew Normal|
  type      =density|
  pdf_image  =[[Image:Skew normal densities.svg|325px|Probability density plots of skew normal distributions]]|
  cdf_image  =[[Image:Skew normal cdfs.svg|325px|Cumulative distribution function plots of skew normal distributions]]|
  parameters =<math>\xi \,</math> [[location parameter|location]] ([[real number|real]])<br/><math>\omega \,</math> [[scale parameter|scale]] (positive, [[real number|real]])<br/><math>\alpha \,</math> [[shape parameter|shape]] ([[real number|real]])|
  support    =<math>x \in (-\infty; +\infty)\!</math>|
  pdf        = <math>\frac{1}{\omega\pi} e^{-\frac{(x-\xi)^2}{2\omega^2}} \int_{-\infty}^{\alpha\left(\frac{x-\xi}{\omega}\right)}  e^{-\frac{t^2}{2}}\ dt</math>|
  cdf        =<math>\Phi\left(\frac{x-\xi}{\omega}\right)-2T\left(\frac{x-\xi}{\omega},\alpha\right)</math><br/><math>T(h,a)</math> is [[Owen's T function]]|
  mean      =<math>\xi + \omega\delta\sqrt{\frac{2}{\pi}}</math> where <math>\delta = \frac{\alpha}{\sqrt{1+\alpha^2}}</math>|
  median    =<!-- to do -->|
  mode      =<!-- to do -->|
  variance  =<math>\omega^2\left(1 - \frac{2\delta^2}{\pi}\right)</math>|
  skewness  =<math>\gamma_3 = \frac{4-\pi}{2} \frac{\left(\delta\sqrt{2/\pi}\right)^3}{  \left(1-2\delta^2/\pi\right)^{3/2}}</math>|
  kurtosis  =<math>2(\pi - 3)\frac{\left(\delta\sqrt{2/\pi}\right)^4}{\left(1-2\delta^2/\pi\right)^2}</math>|
  entropy    =<!-- to do -->|
  mgf        =<math>M_X\left(t\right)=2\exp\left(\xi t+\frac{\omega^2t^2}{2}\right)\Phi\left(\omega\delta t\right)</math>|
  char      =<!-- to do -->|
}}
 
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 <math>\phi(x)</math> denote the standard normal [[probability density function]]
:<math>\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}</math>
with the [[cumulative distribution function]] given by
:<math>\Phi(x) = \int_{-\infty}^{x} \phi(t)\ dt = \frac{1}{2} \left[ 1 + \operatorname{erf} \left(\frac{x}{\sqrt{2}}\right)\right]</math>
 
where '''erf''' is the [[error function]].  Then the probability density function of the skew-normal distribution with parameter α is given by
:<math>f(x) = 2\phi(x)\Phi(\alpha x). \,</math>
 
This distribution was first introduced by O'Hagan and  Leonard (1976).
 
To add [[location parameter|location]] and [[scale parameter|scale]] parameters to this, one makes the usual transform <math>x\rightarrow\frac{x-\xi}{\omega}</math>. One can verify that the normal distribution is recovered when <math>\alpha = 0</math>, and that the absolute value of the [[skewness]] increases as the absolute value of <math>\alpha</math> increases. The distribution is right skewed if <math>\alpha>0</math> and is left skewed if <math>\alpha<0</math>. The probability density function with location <math>\xi</math>, scale <math>\omega</math>, and parameter <math>\alpha</math> becomes
:<math>f(x) = \frac{2}{\omega}\phi\left(\frac{x-\xi}{\omega}\right)\Phi\left(\alpha \left(\frac{x-\xi}{\omega}\right)\right). \,</math>
Note, however, that the skewness of the distribution is limited to the interval <math>(-1,1)</math>.
 
==Estimation==
 
[[Maximum likelihood]] estimates for <math>\xi</math>, <math>\omega</math>, and <math>\alpha</math> can be computed numerically, but no closed-form expression for the estimates is available unless <math>\alpha=0</math>. If a closed-form expression is needed, the [[Method of moments (statistics)|method of moments]] can be applied to estimate <math>\alpha</math> from the sample skew, by inverting the skewness equation. This yields the estimate
 
:<math>|\delta| = \sqrt{\frac{\pi}{2} \frac{  |\hat{\gamma}_3|^{\frac{2}{3}}  }{|\hat{\gamma}_3|^{\frac{2}{3}}+((4-\pi)/2)^\frac{2}{3}}}</math>
 
where <math>\delta = \frac{\alpha}{\sqrt{1+\alpha^2}}</math>, and <math>\hat{\gamma}_3</math> is the sample skew. The sign of <math>\delta</math> is the same as the sign of <math>\hat{\gamma}_3</math>. Consequently, <math>\hat{\alpha} = \delta/\sqrt{1-\delta^2}</math>.
 
The  maximum (theoretical) skewness is obtained by setting <math>{\delta = 1}</math> in the skewness equation, giving <math>\gamma_3 \approx 0.9952717</math>. However it is possible that the sample skewness is larger, and then <math>\alpha</math> cannot be determined from these equations. When using the method of moments in an automatic fashion, for example to give starting values for maximum likelihood iteration, one should therefore let (for example) <math>|\hat{\gamma}_3| = \min(0.99, |(1/n)\sum{((x_i-\bar{x})/s)^3}|)</math>.
 
==See also==
* [[Normal distribution]]
* [[Generalized normal distribution]]
* [[Log-normal distribution]]
* [[Skewness]]
 
==References==
* {{cite journal |last=Azzalini |first=A. |authorlink= |coauthors= |year=1985 |title=A class of distributions which includes the normal ones|journal=Scandinavian Journal of Statistics |volume=12 |issue= |pages=171–178}}
 
* O'Hagan, A. and Leonard, T. (1976). Bayes estimation subject to uncertainty about parameter constraints. Biometrika, 63, 201-202.
 
==External links==
* [http://azzalini.stat.unipd.it/SN/Intro/intro.html A very brief introduction to the skew-normal distribution]
* [http://azzalini.stat.unipd.it/SN/ The Skew-Normal Probability Distribution (and related distributions, such as the skew-t)]
* [http://people.sc.fsu.edu/~burkardt/cpp_src/owens/owens.html OWENS: Owen's T Function]
* [http://dahoiv.net/master/index.html Closed-skew Distributions - Simulation, Inversion and Parameter Estimation]
 
{{ProbDistributions|continuous-infinite}}
{{Statistics|hide}}
 
{{DEFAULTSORT:Skew Normal Distribution}}
[[Category:Continuous distributions]]
[[Category:Probability distributions]]

Revision as of 17:14, 12 February 2014

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