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Gompertz distribution

Probability density function Note: b=2.322
Cumulative distribution function Note: b=2.322
Parameters	${\displaystyle \eta ,b>0\,\!}$
Support	${\displaystyle x\in [0,\infty )\!}$
PDF	${\displaystyle b\eta e^{bx}e^{\eta }\exp \left(-\eta e^{bx}\right)}$
CDF	${\displaystyle 1-\exp \left(-\eta \left(e^{bx}-1\right)\right)}$
Mean	${\displaystyle (-1/b)e^{\eta }{\text{Ei}}\left(-\eta \right)}$ ${\displaystyle {\text{where Ei}}\left(z\right)=\int \limits _{-z}^{\infty }\left(e^{-v}/v\right)dv}$
Mode	${\displaystyle =\left(1/b\right)\ln \left(1/\eta \right)\ }$ ${\displaystyle {\text{with }}0<{\text{F}}\left(x^{*}\right)<1-e^{-1}=0.632121,0<\eta <1}$ ${\displaystyle =0,\quad \eta \geq 1}$
Variance	${\displaystyle \left(1/b\right)^{2}e^{\eta }\{-2\eta {\ }_{3}{\text{F}}_{3}\left(1,1,1;2,2,2;-\eta \right)+\gamma ^{2}}$${\displaystyle +\left(\pi ^{2}/6\right)+2\gamma \ln \left(\eta \right)+^{2}-e^{\eta }^{2}\}}$ ${\displaystyle {\begin{aligned}{\text{ where }}&\gamma {\text{ is the Euler constant: }}\,\!\\&\gamma =-\psi \left(1\right)={\text{0.5777215... }}\end{aligned}}}$${\displaystyle {\begin{aligned}{\text{ and }}{}_{3}{\text{F}}_{3}&\left(1,1,1;2,2,2;-z\right)=\\&\sum _{k=0}^{\infty }\left\left(-1\right)^{k}\left(z^{k}/k!\right)\end{aligned}}}$
MGF	${\displaystyle {\text{E}}\left(e^{-tx}|\eta ,b\right)=\eta e^{\eta }{\text{E}}_{t/b}\left(\eta \right)}$ ${\displaystyle {\text{with E}}_{t/b}\left(\eta \right)=\int _{1}^{\infty }e^{-\eta v}v^{-t/b}dv,\ t>0}$

The Gompertz distribution is an extreme value (reverted Gumbel distribution) distribution (i.e., the distribution of ${\displaystyle -x}$) truncated at zero. It has been used as a model of customer lifetime.

Specification

Probability density function

The probability density function of the Gompertz distribution is:

${\displaystyle f\left(x|\eta ,b\right)=b\eta e^{bx}e^{\eta }\exp \left(-\eta e^{bx}\right){\text{for }}x\geq 0,\,}$

where ${\displaystyle b>0\,\!}$ is the scale parameter and ${\displaystyle \eta >0\,\!}$ is the shape parameter of the Gompertz distribution.

Cumulative distribution function

The cumulative distribution function of the Gompertz distribution is:

${\displaystyle F\left(x|\eta ,b\right)=1-\exp \left(-\eta \left(e^{bx}-1\right)\right)}$

where ${\displaystyle \eta ,b>0,x\geq 0.\,}$.

Moment generating function

The moment generating function is such as: ${\displaystyle {\text{E}}\left(e^{-tx}|\eta ,b\right)=\eta e^{\eta }{\text{E}}_{t/b}\left(\eta \right)}$
With ${\displaystyle {\text{E}}_{t/b}\left(\eta \right)=\int _{1}^{\infty }e^{-\eta v}v^{-t/b}dv,\ t>0}$

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 ${\displaystyle \eta \,\!}$:

• ${\displaystyle \eta \geq 1\,}$ the probability density function has its mode at 0.
• ${\displaystyle \eta <1\,}$ the probability density function has its mode at

${\displaystyle x^{*}=\left(1/b\right)\ln \left(1/\eta \right){\text{with }}0<{\text{F}}\left(x^{*}\right)<1-e^{-1}=0.632121}$

Related distributions

The Gamma distribution is a natural conjugate prior to a Gompertz likelihood with known scale parameter. If ${\displaystyle \eta \,\!}$ varies according to a gamma distribution with shape parameter ${\displaystyle \alpha \,\!}$ and scale parameter ${\displaystyle \beta \,\!}$ (mean = ${\displaystyle \alpha \beta \,\!}$), the cumulative distribution function of ${\displaystyle x}$ is Gamma/Gompertz (G/G).

See also

• Gompertz function
• Gompertz–Makeham law of mortality
• Customer lifetime value
• Gumbel distribution
• Gamma distribution

References

• Bemmaor, Albert C. (2011). "Modeling Purchasing Behavior With Sudden 'Death': A Flexible Customer Lifetime Model". Management Science. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Bemmaor, Albert C.; Glady, Nicolas (2011). "Implementing the Gamma/Gompertz/NBD Model in MATLAB" (PDF). Cergy-Pontoise: ESSEC Business School.
• Gompertz, B. (1825). "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". Philosophical Transactions of the Royal Society of London. 115: 513–583. {{cite journal}}: Cite has empty unknown parameter: |1= (help)
• Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1995). "Continuous Univariate Distributions". 2 (2nd ed.). New York: John Wiley & Sons. {{cite journal}}: Cite has empty unknown parameter: |1= (help); Cite journal requires |journal= (help)
• Sheikh, A. K. (1989). "Truncated Extreme Value Model for Pipeline Reliability". Reliability Engineering and System Safety. 25 (1): 1–14. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Continuous distributions