# Gompertz distribution

In probability and statistics, the Gompertz distribution is a continuous probability distribution, named after Benjamin Gompertz. The Gompertz distribution is often applied to describe the distribution of adult lifespans by demographers[1][2] and actuaries.[3][4] Related fields of science such as biology[5] and gerontology[6] also considered the Gompertz distribution for the analysis of survival. More recently, computer scientists have also started to model the failure rates of computer code by the Gompertz distribution.[7] In Marketing Science, it has been used as an individual-level simulation for customer lifetime value modeling.[8] In network theory, particularly the Erdős–Rényi model, the walk length of a random self-avoiding walk (SAW) is distributed according to the Gompertz distribution.[9]

Parameters Probability density function Cumulative distribution function shape ${\displaystyle \eta >0\,\!}$, scale ${\displaystyle b>0\,\!}$ ${\displaystyle x\in [0,\infty )\!}$ ${\displaystyle b\eta \exp \left(\eta +bx-\eta e^{bx}\right)}$ ${\displaystyle 1-\exp \left(-\eta \left(e^{bx}-1\right)\right)}$ ${\displaystyle (1/b)e^{\eta }{\text{Ei}}\left(-\eta \right)}$${\displaystyle {\text{where Ei}}\left(z\right)=\int \limits _{-z}^{\infty }\left(e^{-v}/v\right)dv}$ ${\displaystyle \left(1/b\right)\ln \left[\left(-1/\eta \right)\ln \left(1/2\right)+1\right]}$ ${\displaystyle =\left(1/b\right)\ln \left(1/\eta \right)\ }$${\displaystyle {\text{with }}0<{\text{F}}\left(x^{*}\right)<1-e^{-1}=0.632121,0<\eta <1}$${\displaystyle =0,\quad \eta \geq 1}$ ${\displaystyle \left(1/b\right)^{2}e^{\eta }\{-2\eta {\ }_{3}{\text{F}}_{3}\left(1,1,1;2,2,2;-\eta \right)+\gamma ^{2}}$${\displaystyle +\left(\pi ^{2}/6\right)+2\gamma \ln \left(\eta \right)+[\ln \left(\eta \right)]^{2}-e^{\eta }[{\text{Ei}}\left(-\eta \right)]^{2}\}}${\displaystyle {\begin{aligned}{\text{ where }}&\gamma {\text{ is the Euler constant: }}\,\!\\&\gamma =-\psi \left(1\right)={\text{0.577215... }}\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[1/\left(k+1\right)^{3}\right]\left(-1\right)^{k}\left(z^{k}/k!\right)\end{aligned}}} ${\displaystyle {\text{E}}\left(e^{-tx}\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}$

## Specification

### Probability density function

The probability density function of the Gompertz distribution is:

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

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

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

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

### Moment generating function

The moment generating function is:

${\displaystyle {\text{E}}\left(e^{-tX}\right)=\eta e^{\eta }{\text{E}}_{t/b}\left(\eta \right)}$

where

${\displaystyle {\text{E}}_{t/b}\left(\eta \right)=\int _{1}^{\infty }e^{-\eta v}v^{-t/b}dv,\ t>0.}$

## Properties

The Gompertz distribution is a flexible distribution that can be skewed to the right and to the left. Its hazard function ${\displaystyle h(x)=\eta be^{bx}}$  is a convex function of ${\displaystyle F\left(x;\eta ,b\right)}$ . The model can be fitted into the innovation-imitation paradigm with ${\displaystyle p=\eta b}$  as the coefficient of innovation and ${\displaystyle b}$  as the coefficient of imitation. When ${\displaystyle t}$  becomes large, ${\displaystyle z(t)}$  approaches ${\displaystyle \infty }$ . The model can also belong to the propensity-to-adopt paradigm with ${\displaystyle \eta }$  as the propensity to adopt and ${\displaystyle b}$  as the overall appeal of the new offering.

### Shapes

The Gompertz density function can take on different shapes depending on the values of the shape parameter ${\displaystyle \eta \,\!}$ :

• When ${\displaystyle \eta \geq 1,\,}$  the probability density function has its mode at 0.
• When ${\displaystyle 0<\eta <1,\,}$  the probability density function has its mode at
${\displaystyle x^{*}=\left(1/b\right)\ln \left(1/\eta \right){\text{with }}0

### Kullback-Leibler divergence

If ${\displaystyle f_{1}}$  and ${\displaystyle f_{2}}$  are the probability density functions of two Gompertz distributions, then their Kullback-Leibler divergence is given by

{\displaystyle {\begin{aligned}D_{KL}(f_{1}\parallel f_{2})&=\int _{0}^{\infty }f_{1}(x;b_{1},\eta _{1})\,\ln {\frac {f_{1}(x;b_{1},\eta _{1})}{f_{2}(x;b_{2},\eta _{2})}}dx\\&=\ln {\frac {e^{\eta _{1}}\,b_{1}\,\eta _{1}}{e^{\eta _{2}}\,b_{2}\,\eta _{2}}}+e^{\eta _{1}}\left[\left({\frac {b_{2}}{b_{1}}}-1\right)\,\operatorname {Ei} (-\eta _{1})+{\frac {\eta _{2}}{\eta _{1}^{\frac {b_{2}}{b_{1}}}}}\,\Gamma \left({\frac {b_{2}}{b_{1}}}+1,\eta _{1}\right)\right]-(\eta _{1}+1)\end{aligned}}}

where ${\displaystyle \operatorname {Ei} (\cdot )}$  denotes the exponential integral and ${\displaystyle \Gamma (\cdot ,\cdot )}$  is the upper incomplete gamma function.[10]

## 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 ${\displaystyle b\,\!.}$ [8]
• When ${\displaystyle \eta \,\!}$  varies according to a gamma distribution with shape parameter ${\displaystyle \alpha \,\!}$  and scale parameter ${\displaystyle \beta \,\!}$  (mean = ${\displaystyle \alpha /\beta \,\!}$ ), the distribution of ${\displaystyle x}$  is Gamma/Gompertz.[8]

Gompertz distribution fitted to maximum monthly 1-day rainfalls [11]

## Notes

1. ^ Vaupel, James W. (1986). "How change in age-specific mortality affects life expectancy" (PDF). Population Studies. 40 (1): 147–157. doi:10.1080/0032472031000141896. PMID 11611920.
2. ^ Preston, Samuel H.; Heuveline, Patrick; Guillot, Michel (2001). Demography:measuring and modeling population processes. Oxford: Blackwell.
3. ^ Benjamin, Bernard; Haycocks, H.W.; Pollard, J. (1980). The Analysis of Mortality and Other Actuarial Statistics. London: Heinemann.
4. ^ Willemse, W. J.; Koppelaar, H. (2000). "Knowledge elicitation of Gompertz' law of mortality". Scandinavian Actuarial Journal. 2000 (2): 168–179. doi:10.1080/034612300750066845.
5. ^ Economos, A. (1982). "Rate of aging, rate of dying and the mechanism of mortality". Archives of Gerontology and Geriatrics. 1 (1): 46–51. doi:10.1016/0167-4943(82)90003-6. PMID 6821142.
6. ^ Brown, K.; Forbes, W. (1974). "A mathematical model of aging processes". Journal of Gerontology. 29 (1): 46–51. doi:10.1093/geronj/29.1.46. PMID 4809664.
7. ^ Ohishi, K.; Okamura, H.; Dohi, T. (2009). "Gompertz software reliability model: estimation algorithm and empirical validation". Journal of Systems and Software. 82 (3): 535–543. doi:10.1016/j.jss.2008.11.840.
8. ^ a b c Bemmaor, Albert C.; Glady, Nicolas (2012). "Modeling Purchasing Behavior With Sudden 'Death': A Flexible Customer Lifetime Model". Management Science. 58 (5): 1012–1021. doi:10.1287/mnsc.1110.1461.
9. ^ Tishby, Biham, Katzav (2016), The distribution of path lengths of self avoiding walks on Erdős-Rényi networks, arXiv:1603.06613.
10. ^ Bauckhage, C. (2014), Characterizations and Kullback-Leibler Divergence of Gompertz Distributions, arXiv:1402.3193.
11. ^ Calculator for probability distribution fitting [1]