# Gompertz distribution

Short description: Continuous probability distribution, named after Benjamin Gompertz
Parameters Probability density function Cumulative distribution function shape $\displaystyle{ \eta\gt 0\,\! }$, scale $\displaystyle{ b \gt 0\,\! }$ $\displaystyle{ x \in [0, \infty)\! }$ $\displaystyle{ b\eta \exp\left(\eta + bx -\eta e^{bx} \right) }$ $\displaystyle{ 1-\exp\left(-\eta\left(e^{bx}-1 \right)\right) }$ $\displaystyle{ (1/b)e^{\eta}\text{Ei}\left(-\eta\right) }$$\displaystyle{ \text {where Ei}\left(z\right)=\int\limits_{-z}^{\infin}\left(e^{-v}/v\right)dv }$ $\displaystyle{ \left(1/b\right)\ln\left[\left(1/\eta\right)\ln\left(1/2\right)+1\right] }$ $\displaystyle{ =\left(1/b\right)\ln \left(1/\eta\right)\ }$$\displaystyle{ \text {with }0 \lt \text {F}\left(x^*\right)\lt 1-e^{-1} = 0.632121, 0\lt \eta\lt 1 }$$\displaystyle{ =0, \quad \eta \ge 1 }$ $\displaystyle{ \left(1/b\right)^2 e^{\eta}\{-2\eta { \ }_3\text {F}_3 \left(1,1,1;2,2,2;\eta\right)+\gamma^2 }$$\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{align}\text{ where } &\gamma \text{ is the Euler constant: }\,\!\\ &\gamma=-\psi\left(1\right)=\text{0.577215... }\end{align} }\displaystyle{ \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} } $\displaystyle{ \text{E}\left(e^{-t x}\right)=\eta e^{\eta}\text{E}_{t/b}\left(\eta\right) }$$\displaystyle{ \text{with E}_{t/b}\left(\eta\right)=\int_1^\infin e^{-\eta v} v^{-t/b}dv,\ t\gt 0 }$

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]

## Specification

### Probability density function

The probability density function of the Gompertz distribution is:

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

where $\displaystyle{ b \gt 0\,\! }$ is the scale parameter and $\displaystyle{ \eta \gt 0\,\! }$ 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 $\displaystyle{ \eta, b\gt 0, }$ and $\displaystyle{ x \geq 0 \, . }$

### Moment generating function

The moment generating function is:

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

where

$\displaystyle{ \text{E}_{t/b}\left(\eta\right)=\int_1^\infin e^{-\eta v} v^{-t/b}dv,\ t\gt 0. }$

## Properties

The Gompertz distribution is a flexible distribution that can be skewed to the right and to the left. Its hazard function $\displaystyle{ h(x)=\eta b e^{bx} }$ 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 $\displaystyle{ 0 \lt \eta \lt 1,\, }$ the probability density function has its mode at
$\displaystyle{ x^*=\left(1/b\right)\ln \left(1/\eta\right)\text {with }0 \lt F\left(x^*\right)\lt 1-e^{-1} = 0.632121 }$

### 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{align} 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{align} }

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]
• If $\displaystyle{ Y \sim \mathrm{Gompertz} }$, then $\displaystyle{ X = \exp(Y) \sim \mathrm{Weibull}^{-1} }$, and hence $\displaystyle{ \exp(-Y) \sim \mathrm{Weibull} }$.[12]

## Applications

• In hydrology the Gompertz distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. The blue picture illustrates an example of fitting the Gompertz distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

## Notes

1. Vaupel, James W. (1986). "How change in age-specific mortality affects life expectancy". 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. 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]
12. Kleiber, Christian; Kotz, Samuel (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Wiley. p. 179. doi:10.1002/0471457175. ISBN 9780471150640.