|
|
| Line 1: |
Line 1: |
| {{Probability distribution|
| | In [[physics]], the term '''total pressure''' may indicate two different quantities, both having the dimensions of a [[pressure]]: |
| name = Shifted Gompertz|
| | * In [[fluid dynamics]], ''total pressure'' (<math>p_0</math>) refers to the sum of [[static pressure]] p, [[dynamic pressure]] q, and gravitational head, as expressed by [[Bernoulli's_principle#Incompressible_flow_equation|Bernoulli's principle]]: |
| type =density|
| | :<math>p_0 = p + q + \rho g z\,</math> |
| pdf_image =[[File:Shiftedgompertz distribution PDF new.png|325px|Probability density plots of shifted Gompertz distributions]]|
| | :where ρ is the density of the fluid, g is the local acceleration due to gravity, and z is the height above a datum. |
| cdf_image =[[File:Shiftedgompertz distribution CDF new.png|325px|Cumulative distribution plots of shifted Gompertz distributions]]|
| | :If the variation in height above the datum is zero, or so small it can be ignored, the above equation reduces to the following simplified form: |
| parameters =<math>b>0</math> [[scale parameter|scale]] ([[real number|real]])<br/><math>\eta>0</math> [[shape parameter|shape]] (real)|
| | :<math>p_0 = p + q\,</math> |
| 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 ==
| | * In a mixture of [[ideal gas]]es, ''total pressure'' refers to the sum of each gas' [[partial pressure]]. |
| ===Probability density function===
| |
|
| |
|
| The [[probability density function]] of the shifted Gompertz distribution is:
| | [[Category:Pressure]] |
| | | [[Category:Fluid dynamics]] |
| :<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>
| | [[Category:Gases]] |
| | |
| | |
| where <math>b > 0</math> is the [[scale parameter]] and <math>\eta > 0</math> is the [[shape parameter]] of the shifted Gompertz distribution.
| |
| | |
| ===Cumulative distribution function===
| |
| | |
| The [[cumulative distribution function]] of the shifted Gompertz distribution is:
| |
| | |
| :<math> F(x;b,\eta) = \left(1 - e^{-bx}\right)e^{-\eta e^{-bx}} \text{ for }x \geq 0. \,</math> | |
| | |
| | |
| == Properties ==
| |
| The shifted Gompertz distribution is right-skewed for all values of <math>\eta</math>. It is more flexible than the [[Gumbel distribution]].
| |
| | |
| ===Shapes===
| |
| 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 ==
| |
| 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]].
| |
| | |
| == See also ==
| |
| *[[Gumbel distribution]]
| |
| *[[Generalized extreme value distribution]]
| |
| *[[Mixture model]]
| |
| *[[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]] | |
