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Transvections

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2011-04-29T03:23:30Z

Reverted edits by 72.70.145.171 (talk) to last revision by CBM (HG)

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{{more footnotes|date=December 2011}}
{{Probability distribution
| name = Gompertz distribution
| type = density
| pdf_image =[[File:Gompertz distrbution.png|325px|Gompertz distribution]] Note: b=2.322
| cdf_image =[[File:Gompertz cum dist nokey.png|325px|Gompertz cumulative distribution]]
| parameters =<math>\eta, b > 0\,\!</math>
| support =<math>x \in [0, \infty)\!</math>
| pdf =<math>b\eta e^{bx}e^{\eta}\exp\left(-\eta e^{bx} \right)</math>
| cdf =<math>1-\exp\left(-\eta\left(e^{bx}-1 \right)\right)</math>
| mean =<math>(-1/b)e^{\eta}\text{Ei}\left(-\eta\right)</math> <math> \text {where Ei}\left(z\right)=\int\limits_{-z}^{\infin}\left(e^{-v}/v\right)dv</math>
| median =<math>\left(1/b\right)\ln\left[\left(-1/\eta\right)\ln\left(1/2\right)+1\right]</math>
| mode = <math>=\left(1/b\right)\ln \left(1/\eta\right)\ </math> <math>\text {with }0 <\text {F}\left(x^*\right)<1-e^{-1} = 0.632121, 0<\eta<1 </math> <math>=0, \quad \eta \ge 1</math>
| variance =<math>\left(1/b\right)^2 e^{\eta}\{-2\eta { \ }_3\text {F}_3 \left(1,1,1;2,2,2;-\eta\right)+\gamma^2</math><math>
+\left(\pi^2/6\right)+2\gamma\ln\left(\eta\right)+[\ln\left(\eta\right)]^2-e^{\eta}[\text{Ei}\left(-\eta \right)]^2\}</math> <math>\begin{align}\text{ where } &\gamma \text{ is the Euler constant: }\,\!\\ &\gamma=-\psi\left(1\right)=\text{0.5777215... }\end{align}</math><math>\begin{align}\text { and } { }_3\text {F}_3&\left(1,1,1;2,2,2;-z\right)=\\&\sum_{k=0}^\infty\left[1/\left(k+1\right)^3\right]\left(-1\right)^k\left(z^k/k!\right)\end{align}</math>
| skewness =
| kurtosis =
| entropy =
| mgf =<math>\text{E}\left(e^{-t x}\right)=\eta e^{\eta}\text{E}_{t/b}\left(\eta\right)</math> <math>\text{with E}_{t/b}\left(\eta\right)=\int_1^\infin e^{-\eta v} v^{-t/b}dv,\ t>0</math>
| char =
}}

In [[probability and statistics]], the '''Gompertz distribution''' is a [[continuous probability distribution]]. The Gompertz distribution is often applied to describe the distribution of adult lifespans by [[demographer]]s<ref name=Vaupel1986/><ref name=Preston2001/> and [[actuaries]].<ref name=Benjamin1980/><ref name=Willemse2000/> Related fields of science such as biology<ref name=Economos1982/> and gerontology<ref name=Brown1974/> also considered the Gompertz distribution for the analysis of survival. More recently, computer scientists have also started to model the failure rates of computer codes by the Gompertz distribution.<ref name=Ohishi2009/> In marketing science, it has been used as an individual-level model of customer lifetime.<ref name=BG/>

== Specification ==

===Probability density function===

The [[probability density function]] of the Gompertz distribution is:

:<math>f\left(x;\eta, b\right)=b\eta e^{bx}e^{\eta}\exp\left(-\eta e^{bx} \right)\text{for }x \geq 0, \,</math>

where <math>b > 0\,\!</math> is the [[scale parameter]] and <math>\eta > 0\,\!</math> is the [[shape parameter]] of the Gompertz distribution. In the actuarial and biological sciences and in demography, the Gompertz distribution is parametrized slightly differently ([[Gompertz–Makeham law of mortality]]).

===Cumulative distribution function===

The [[cumulative distribution function]] of the Gompertz distribution is:

:<math>F\left(x;\eta, b\right)= 1-\exp\left(-\eta\left(e^{bx}-1 \right)\right) ,</math>
where <math>\eta, b>0,</math> and <math> x \geq 0 \, .</math>

===Moment generating function===
The moment generating function is:
:<math>\text{E}\left(e^{-t X}\right)=\eta e^{\eta}\text{E}_{t/b}\left(\eta\right)</math>
where
:<math>\text{E}_{t/b}\left(\eta\right)=\int_1^\infin e^{-\eta v} v^{-t/b}dv,\ t>0.</math>

== Properties ==
The Gompertz distribution is a flexible distribution that can be skewed to the right and to the left.

===Shapes===
The Gompertz density function can take on different shapes depending on the values of the shape parameter <math>\eta\,\!</math>:
* When <math>\eta \geq 1,\,</math> the probability density function has its mode at 0.
* When <math>0 < \eta < 1,\,</math> the probability density function has its mode at
::<math>x^*=\left(1/b\right)\ln \left(1/\eta\right)\text {with }0 < F\left(x^*\right)<1-e^{-1} = 0.632121</math>

== Related distributions ==
*If ''X'' is defined to be the result of sampling from a [[Gumbel distribution]] until a negative value ''Y'' is produced, and setting ''X''=−''Y'', then ''X'' has a Gompertz distribution.
*The [[gamma distribution]] is a natural [[conjugate prior]] to a Gompertz likelihood with known scale parameter <math>b \,\!.</math><ref name=BG/>
* When <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/Gompertz.<ref name=BG/>

== See also ==
*[[Gompertz function]]
*[[Customer lifetime value]]
*[[Gamma Gompertz distribution]]

==Notes==
{{Reflist|refs=

<ref name=BG>{{cite journal
| last=Bemmaor | first=Albert C. | coauthors=Glady, Nicolas
| title=Modeling Purchasing Behavior With Sudden 'Death': A Flexible Customer Lifetime Model
| volume = 58 | issue=5 | pages = 1012–1021
| journal=Management Science | year=2012 | doi=10.1287/mnsc.1110.1461}}</ref>
<ref name=Vaupel1986>{{cite journal
|last=Vaupel | first=James W.
|title=How change in age-specific mortality affects life expectancy
|volume=40 | issue=1 | pages=147–157
|journal=Population Studies
|year=1986}}</ref>
<ref name=Preston2001>{{cite book
|last=Preston | first=Samuel H.
|coauthors=Heuveline, Patrick and Guillot, Michel
|title=Demography:measuring and modeling population processes
|publisher=Blackwell | location=Oxford
|year=2001}}</ref>
<ref name=Benjamin1980>{{cite book
|last=Benjamin | first=Bernard
|coauthors=Haycocks, H.W. and Pollard, J.
|title=The Analysis of Mortality and Other Actuarial Statistics
|publisher=Heinemann | location=London
|year=1980}}</ref>
<ref name=Willemse2000>{{cite journal
|last=Willemse | first=W. J. | coauthors=Koppelaar, H.
|title=Knowledge elicitation of Gompertz' law of mortality
|journal=Scandinavian Actuarial Journal |issue=2 | pages=168–179
|year=2000}}</ref>
<ref name=Brown1974>{{cite journal
|last=Brown | first=K. | coauthors=Forbes, W.
|title=A mathematical model of aging processes
|journal=Journal of Gerontology |volume=29 | issue=1 | pages=46–51
|year=1974}}</ref>
<ref name=Economos1982>{{cite journal
|last=Economos | first=A.
|title=Rate of aging, rate of dying and the mechanism of mortality
|journal=Archives of Gerontology and Geriatrics
|volume=1 | issue=1 | pages=46–51
|year=1982}}</ref>
<ref name=Ohishi2009>{{cite journal
|last=Ohishi |first=K. |coauthors=Okamura, H. and Dohi, T.
|title=Gompertz software reliability model: estimation algorithm and empirical validation
|journal=Journal of Systems and Software
|volume=82 | issue=3 | pages=535–543
|year=2009}}</ref>
}}

==References==
*{{cite web|last=Bemmaor|first=Albert C. |last2=Glady|first2=Nicolas| year = 2011 |title=Implementing the Gamma/Gompertz/NBD Model in MATLAB|url=http://dl.dropbox.com/u/7097708/gg_nbd_MATLAB.pdf| publisher = ESSEC Business School|location = Cergy-Pontoise }}
*{{Cite journal | last1=Gompertz | first=B. | year= 1825 |pages=513–583| title=On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies | journal =Philosophical Transactions of the Royal Society of London|
volume = 115 |jstor=107756 | doi=10.1098/rstl.1825.0026}}
*{{Cite journal
| last=Johnson | first=Norman L. | last2=Kotz | first2=Samuel
| last3=Balakrishnan | first3=N. | year= 1995
| title=Continuous Univariate Distributions | volume=2
| edition=2nd | publisher=John Wiley & Sons | location=New York
| isbn=0-471-58494-0 | pages=25–26}}
*{{cite journal|last=Sheikh|first=A. K.|coauthors=Boah, J. K.; Younas, M. |title=Truncated Extreme Value Model for Pipeline Reliability|journal=Reliability Engineering and System Safety|year=1989|volume = 25|issue=1|pages=1–14 |doi=10.1016/0951-8320(89)90020-3}}

{{ProbDistributions|continuous-semi-infinite}}

[[Category:Continuous distributions]]
[[Category:Probability distributions]]

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en>Jiahuang: /* Transvections */

en>Σ: Reverted edits by 72.70.145.171 (talk) to last revision by CBM (HG)