Gompertz distribution

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Gompertz distribution
Probability density function
Gompertz distribution
Note: b=2.322
Cumulative distribution function
Gompertz cumulative distribution
Parameters \eta, b > 0\,\!
Support x \in [0, \infty)\!
pdf b\eta e^{bx}e^{\eta}\exp\left(-\eta e^{bx} \right)
CDF 1-\exp\left(-\eta\left(e^{bx}-1 \right)\right)
Mean (-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
Median \left(1/b\right)\ln\left[\left(-1/\eta\right)\ln\left(1/2\right)+1\right]
Mode =\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
Variance \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}
MGF \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. Early users in the 1990s for the Gompertz distribution in CLV models included Edge Consulting and BrandScience[8]

Specification[edit]

Probability density function[edit]

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[edit]

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[edit]

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[edit]

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

Shapes[edit]

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

Related distributions[edit]

  • 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]

See also[edit]

Notes[edit]

  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. 

References[edit]

  • Bemmaor, Albert C.; Glady, Nicolas (2011). "Implementing the Gamma/Gompertz/NBD Model in MATLAB". 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.