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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 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 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]

Gompertz distribution
Probability density function
GompertzPDF.svg
Cumulative distribution function
GompertzCDF.svg
Parameters
Support
PDF
CDF
Mean
Median
Mode

Variance
MGF

Contents

SpecificationEdit

Probability density functionEdit

The probability density function of the Gompertz distribution is:

 

where   is the scale parameter and   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 functionEdit

The cumulative distribution function of the Gompertz distribution is:

 

where   and  

Moment generating functionEdit

The moment generating function is:

 

where

 

PropertiesEdit

The Gompertz distribution is a flexible distribution that can be skewed to the right and to the left. Its hazard function is a convex function of  . The model can be fitted into the innovation-imitation paradigm with   as the coefficient of innovation and   as the coefficient of imitation. When   becomes large,   approaches  . The model can also belong to the propensity-to-adopt paradigm with   as the propensity to adopt and   as the overall appeal of the new offering.

ShapesEdit

The Gompertz density function can take on different shapes depending on the values of the shape parameter  :

  • When   the probability density function has its mode at 0.
  • When   the probability density function has its mode at
 

Kullback-Leibler divergenceEdit

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

 

where   denotes the exponential integral and   is the upper incomplete gamma function.[10]

Related distributionsEdit

  • 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  [8]
  • When   varies according to a gamma distribution with shape parameter   and scale parameter   (mean =  ), the distribution of   is Gamma/Gompertz.[8]
 
Gompertz distribution fitted to maximum monthly 1-day rainfalls [11]

ApplicationsEdit

See alsoEdit

NotesEdit

  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.
  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.
  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.; 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]

ReferencesEdit

  • 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.