Gompertz distribution

Gompertz distribution
	Probability density function
	Cumulative distribution function
Parameters	shape , scale
Support
PDF
CDF
Mean	;
Median
Mode	; ;
Variance	;
MGF	;

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]

Specification

Probability density function

The probability density function of the Gompertz distribution is:

f\left(x;\eta ,b\right)=b\eta \exp \left(\eta +bx-\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^{-tX}\right)=\eta e^{\eta }{\text{E}}_{t/b}\left(\eta \right)

where

{\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 $h(x)=\eta be^{bx}$ is a convex function of $F\left(x;\eta ,b\right)$ . The model can be fitted into the innovation-imitation paradigm with $p=\eta b$ as the coefficient of innovation and $b$ as the coefficient of imitation. When $t$ becomes large, $z(t)$ approaches $\infty$ . The model can also belong to the propensity-to-adopt paradigm with $\eta$ as the propensity to adopt and $b$ 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 $\eta \,\!$ :

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{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 $\operatorname {Ei} (\cdot )$ denotes the exponential integral and $\Gamma (\cdot ,\cdot )$ 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 $b\,\!.$ [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]

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

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

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.
Preston, Samuel H.; Heuveline, Patrick; Guillot, Michel (2001). Demography:measuring and modeling population processes. Oxford: Blackwell.
Benjamin, Bernard; Haycocks, H.W.; Pollard, J. (1980). The Analysis of Mortality and Other Actuarial Statistics. London: Heinemann.
Willemse, W. J.; Koppelaar, H. (2000). "Knowledge elicitation of Gompertz' law of mortality". Scandinavian Actuarial Journal. 2000 (2): 168–179. doi:10.1080/034612300750066845.
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.
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.
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.
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.
Tishby, Biham, Katzav (2016), The distribution of path lengths of self avoiding walks on Erdős-Rényi networks, arXiv:1603.06613.
Bauckhage, C. (2014), Characterizations and Kullback-Leibler Divergence of Gompertz Distributions, arXiv:1402.3193.
Calculator for probability distribution fitting

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.

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[Vaupel1986-1] 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.

[Preston2001-2] Preston, Samuel H.; Heuveline, Patrick; Guillot, Michel (2001). Demography:measuring and modeling population processes. Oxford: Blackwell.

[Benjamin1980-3] Benjamin, Bernard; Haycocks, H.W.; Pollard, J. (1980). The Analysis of Mortality and Other Actuarial Statistics. London: Heinemann.

[Willemse2000-4] Willemse, W. J.; Koppelaar, H. (2000). "Knowledge elicitation of Gompertz' law of mortality". Scandinavian Actuarial Journal. 2000 (2): 168–179. doi:10.1080/034612300750066845.

[Economos1982-5] 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.

[Brown1974-6] 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.

[Ohishi2009-7] 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.

[BG-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

Probability distributions (List)
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Fréchet gamma gamma/Gompertz generalized gamma generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared noncentral F Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

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