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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      = Shifted Gompertz|
  type      =density|
  pdf_image  =[[File:Shiftedgompertz distribution PDF new.png|325px|Probability density plots of shifted Gompertz distributions]]|
  cdf_image  =[[File:Shiftedgompertz distribution CDF new.png|325px|Cumulative distribution plots of shifted Gompertz distributions]]|
  parameters =<math>b>0</math> [[scale parameter|scale]] ([[real number|real]])<br/><math>\eta>0</math> [[shape parameter|shape]] (real)|
  support    =<math>x \in [0, \infty)\!</math>|
  pdf        =<math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>|
  cdf        =<math>\left(1 - e^{-bx}\right)e^{-\eta e^{-bx}}</math>|
  mean      =<math>(-1/b)\{\mathrm{E}[\ln(X)] - \ln(\eta)\}\,</math>
where <math>X = \eta e^{-bx}\,</math> and
<math>\begin{align}\mathrm{E}[\ln(X)] =& [1 {+} 1 / \eta]\!\!\int_0^\eta \!\!\!\! e^{-X}[\ln(X)]dX\\ &- 1/\eta\!\! \int_0^\eta \!\!\!\! X e^{-X}[\ln(X)] dX \end{align}</math>|
  median    =|
  mode      = <math>0 \text{ for }0 < \eta \leq 0.5</math><br/> <math>(-1/b)\ln(z^\star)\text{, for } \eta > 0.5</math><br/><math>\text{ where }z^\star = [3 + \eta - (\eta^2 + 2\eta + 5)^{1/2}]/(2\eta)</math>|
  variance  =<math>(1/b^2)(\mathrm{E}\{[\ln(X)]^2\} - (\mathrm{E}[\ln(X)])^2)\,</math>
where <math>X = \eta e^{-bx}\,</math> and <math>\begin{align}\mathrm{E}\{[\ln(X)]^2\} =& [1 {+} 1 / \eta]\!\!\int_0^\eta \!\!\!\! e^{-X}[\ln(X)]^2 dX\\ &- 1/\eta \!\!\int_0^\eta \!\!\!\! X e^{-X}[\ln(X)]^2 dX \end{align}</math>|
  skewness  =|
  kurtosis  =|
  entropy    =|
  mgf        =|
  char      =|
}}
The '''shifted Gompertz distribution''' is the distribution of the largest of two independent [[random variable]]s one of which has an [[exponential distribution]] with parameter b and the other has a [[Gumbel distribution]] with parameters <math>\eta</math> and b. In its original formulation the distribution was expressed referring to the Gompertz distribution instead of the Gumbel distribution but, since the Gompertz distribution is a reverted Gumbel distribution ([[truncated distribution|truncated]] at zero), the labelling can be considered as accurate. It has been used as a model of [[Diffusion of innovations|adoption of innovations]]. It was proposed by Bemmaor (1994).  


== Specification ==
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]]
===Probability density function===
* 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.


The [[probability density function]] of the shifted Gompertz distribution is:
Registered users will be able to choose between the following three rendering modes:  


:<math> f(x;b,\eta) = b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right] \text{ for }x \geq 0. \,</math>
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


where <math>b > 0</math> is the [[scale parameter]] and <math>\eta > 0</math> is the [[shape parameter]] of the shifted Gompertz distribution.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


===Cumulative distribution function===
<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].


The [[cumulative distribution function]] of the shifted Gompertz distribution is:
==Demos==


:<math> F(x;b,\eta) = \left(1 - e^{-bx}\right)e^{-\eta e^{-bx}} \text{ for }x \geq 0. \,</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:




== Properties ==
* accessibility:
The shifted Gompertz distribution is right-skewed for all values of <math>\eta</math>. It is more flexible than the [[Gumbel distribution]].  
** 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.


===Shapes===
==Test pages ==
The shifted Gompertz density function can take on different shapes depending on the values of the shape parameter <math>\eta</math>:
* <math>0 < \eta \leq 0.5\,</math> the probability density function has its mode at 0.
* <math>\eta > 0.5\,</math> the probability density function has its mode at
::<math>\text{mode}=-\frac{\ln(z^\star)}{b}\, \qquad 0 < z^\star < 1</math>
:where <math>z^\star\,</math> is the smallest root of
::<math>\eta^2z^2 - \eta(3 + \eta)z + \eta + 1 = 0\,,</math>
:which is
::<math>z^\star = [3 + \eta - (\eta^2 + 2\eta + 5)^{1/2}]/(2\eta).</math>


== Related distributions ==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
If <math>\eta</math> varies according to a [[gamma distribution]] with shape parameter <math>\alpha</math> and scale parameter <math>\beta</math> (mean = <math>\alpha\beta</math>), the distribution of <math>x</math> is Gamma/Shifted Gompertz (G/SG). When <math>\alpha</math> is equal to one, the G/SG reduces to the [[Bass model]].
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


== See also ==
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Gumbel distribution]]
*[[Url2Image|Url2Image (private Wikis only)]]
*[[Generalized extreme value distribution]]
==Bug reporting==
*[[Mixture model]]
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 .
*[[Bass model]]
*[[Gompertz distribution]]
 
== References ==
{{No footnotes|date=April 2012}}
*{{Cite book | surname=Bemmaor | given=Albert C. | year= 1994 |pages=201–223| chapter=Modeling the Diffusion of New Durable Goods: Word-of-Mouth Effect Versus Consumer Heterogeneity | editor=G. Laurent, G.L. Lilien & B. Pras | title=Research Traditions in Marketing | publisher=Kluwer Academic Publishers | place=Boston| ISBN=0-7923-9388-0}}
*{{Cite book| surname1=Chandrasekaran| given1=Deepa | surname2=Tellis| given2=Gerard J. |year= 2007 |volume=3|  chapter=A Critical Review of Marketing Research on Diffusion of New Products | editor=Naresh K. Malhotra  | title=Review of Marketing Research | publisher=M.E. Sharpe | place=Armonk | pages = 39–80 | ISBN = 978-0-7656-1306-6}}
*{{Cite journal
  | last1 = Dover
  | first1 = Yaniv
  | title = Network Traces on Penetration: Uncovering Degree Distribution From Adoption Data 
  | first2 = Jacob |last2=Goldenberg
  | first3 = Daniel |last3=Shapira
  | journal = Marketing Science|doi=10.1287/mksc.1120.0711 |year=2012 }}
*{{Cite journal
  | last1 = Jimenez
  | first1 = Fernando
  | title = A Note on the Moments and Computer Generation of the Shifted Gompertz Distribution 
  | first2 = Pedro |last2=Jodra
  | journal = Communications in Statistics - Theory and Methods
  | volume = 38  | issue = 1
  | pages = 78–89
  | year = 2009
  | doi=10.1080/03610920802155502}}
*{{Cite journal
  | last1 = Van den Bulte
  | first1 = Christophe
  | title = Social Contagion and Income Heterogeneity in New Product Diffusion: A Meta-Analytic Test
  | first2 = Stefan |last2=Stremersch
  | journal = Marketing Science
  | volume = 23  | issue = 4
  | pages = 530–544
  | year = 2004
  | doi = 10.1287/mksc.1040.0054}}
 
{{ProbDistributions|continuous-semi-infinite}}
 
[[Category:Continuous 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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