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{{Probability distribution |
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
  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_1 = \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>|
  cf        =<math>M_X\left(i\delta\omega 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]].
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


==Definition==
Registered users will be able to choose between the following three rendering modes:


Let <math>\phi(x)</math> denote the [[Normal distribution|standard normal]] [[probability density function]]
'''MathML'''
:<math>\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}</math>
:<math forcemathmode="mathml">E=mc^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 (pdf) of the skew-normal distribution with parameter <math>\alpha</math> is given by
<!--'''PNG'''  (currently default in production)
:<math>f(x) = 2\phi(x)\Phi(\alpha x). \,</math>
:<math forcemathmode="png">E=mc^2</math>


This distribution was first introduced by O'Hagan and  Leonard (1976).
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


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.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


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
==Demos==
:<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==
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


[[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>
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


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>.
==Test pages ==


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>.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


==Differential equation==
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
The [[differential equation]] leading to the pdf of the skew normal distribution is
==Bug reporting==
:<math>
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
\omega^4 f''(x)+\left(\alpha^2+2\right) \omega^2 (x-\zeta)
  f'(x)+f(x) \left(\left(\alpha^2+1\right) (x-\zeta )^2+\omega^2\right)=0
</math>,
with initial conditions
:<math>
\begin{array}{l}
\displaystyle f(0)=\frac{\exp\left(-\frac{\zeta^2}{2\omega^2}\right)
  \operatorname{erfc}\left(\frac{\alpha\zeta}{\sqrt{2} \omega}\right)}
  {\sqrt{2\pi}\omega} \text{ and} \\[16pt]
\displaystyle f'(0)=\frac{\exp\left(-\frac{\left(\alpha^2+1\right)\zeta ^2}
  {2 \omega^2}\right)
  \left(2\alpha\omega+\sqrt{2\pi} \zeta
  \exp\left(\frac{\alpha^2 \zeta^2}{2 \omega^2}\right)
  \operatorname{erfc}\left(\frac{\alpha\zeta}{\sqrt{2} \omega}\right)\right)}
  {2\pi\omega^3}.
\end{array}
</math>
 
==See also==
 
* [[Generalized normal distribution]]
* [[Log-normal distribution]]
 
==References==
 
*Andel, J., Netuka, I. and Zvara, K. (1984). On threshold autoregressive processes. Kybernetika, 20, 89-106.
 
* {{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}}
 
* Chan, K-S. and Tong, H. (1986). A note on certain integral equations associated with non-linear time series analysis. Probability and Related Fields, 73, 153-158.
 
* 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]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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