# Gompertz distribution

Parameters Probability density function Note: b=2.322 Cumulative distribution function $\eta, b > 0\,\!$ $x \in [0, \infty)\!$ $b\eta e^{bx}e^{\eta}\exp\left(-\eta e^{bx} \right)$ $1-\exp\left(-\eta\left(e^{bx}-1 \right)\right)$ $(1/b)e^{\eta}\text{Ei}\left(-\eta\right)$ $\text {where Ei}\left(z\right)=\int\limits_{-z}^{\infin}\left(e^{-v}/v\right)dv$ $\left(1/b\right)\ln\left[\left(-1/\eta\right)\ln\left(1/2\right)+1\right]$ $=\left(1/b\right)\ln \left(1/\eta\right)\$ $\text {with }0 <\text {F}\left(x^*\right)<1-e^{-1} = 0.632121, 0<\eta<1$ $=0, \quad \eta \ge 1$ $\left(1/b\right)^2 e^{\eta}\{-2\eta { \ }_3\text {F}_3 \left(1,1,1;2,2,2;-\eta\right)+\gamma^2$$+\left(\pi^2/6\right)+2\gamma\ln\left(\eta\right)+[\ln\left(\eta\right)]^2-e^{\eta}[\text{Ei}\left(-\eta \right)]^2\}$ \begin{align}\text{ where } &\gamma \text{ is the Euler constant: }\,\!\\ &\gamma=-\psi\left(1\right)=\text{0.5777215... }\end{align}\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} $\text{E}\left(e^{-t x}\right)=\eta e^{\eta}\text{E}_{t/b}\left(\eta\right)$ $\text{with E}_{t/b}\left(\eta\right)=\int_1^\infin e^{-\eta v} v^{-t/b}dv,\ t>0$

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 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 codes by the Gompertz distribution.[7] In Marketing Science, it has been used as an individual-level simulation for customer lifetime value modeling.[8] In an independent effort, early users in the 1990s for the Gompertz distribution in CLV models included Edge Consulting and BrandScience

## Specification

### Probability density function

The probability density function of the Gompertz distribution is:

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

where $b > 0\,\!$ is the scale parameter and $\eta > 0\,\!$ 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:

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

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

### Moment generating function

The moment generating function is:

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

where

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

## 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 $\eta\,\!$:

• When $\eta \geq 1,\,$ the probability density function has its mode at 0.
• When $0 < \eta < 1,\,$ the probability density function has its mode at
$x^*=\left(1/b\right)\ln \left(1/\eta\right)\text {with }0 < F\left(x^*\right)<1-e^{-1} = 0.632121$

### Kullback-Leibler divergence

If $f_1$ and $f_2$ are the probability density functions of two Gompertz distributions, then their Kullback-Leibler divergence is given by

\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 $\operatorname{Ei}(\cdot)$ denotes the exponential integral and $\Gamma(\cdot,\cdot)$ is the upper incomplete gamma function.[9]

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

## 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.
2. ^ Preston, Samuel H.; Heuveline, Patrick and Guillot, Michel (2001). Demography:measuring and modeling population processes. Oxford: Blackwell.
3. ^ Benjamin, Bernard; Haycocks, H.W. and 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 (2): 168–179.
5. ^ Economos, A. (1982). "Rate of aging, rate of dying and the mechanism of mortality". Archives of Gerontology and Geriatrics 1 (1): 46–51.
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.
7. ^ Ohishi, K.; Okamura, H. and 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. ^ Bauckhage, C. (2014), Characterizations and Kullback-Leibler Divergence of Gompertz Distributions, arXiv:1402.3193.

## References

• 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. doi:10.1098/rstl.1825.0026. JSTOR 107756.
• Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1995). "Continuous Univariate Distributions" 2 (2nd ed.). New York: John Wiley & Sons. pp. 25–26. ISBN 0-471-58494-0.
• Sheikh, A. K.; Boah, J. K.; Younas, M. (1989). "Truncated Extreme Value Model for Pipeline Reliability". Reliability Engineering and System Safety 25 (1): 1–14. doi:10.1016/0951-8320(89)90020-3.