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In probability and statistics, the Gamma/Gompertz distribution is a continuous probability distribution. It has been used as an aggregate-level model of customer lifetime and a model of mortality risks.

Specification

The probability density function of the Gamma/Gompertz distribution is:

f(x;b,s,\beta )={\frac {bse^{bx}\beta ^{s}}{\left(\beta -1+e^{bx}\right)^{s+1}}}

where $b>0$ is the scale parameter and $\beta ,s>0\,\!$ are the shape parameters of the Gamma/Gompertz distribution.

The cumulative distribution function of the Gamma/Gompertz distribution is:

{\begin{aligned}F(x;b,s,\beta )&=1-{\frac {\beta ^{s}}{\left(\beta -1+e^{bx}\right)^{s}}},{\ }x>0,{\ }b,s,\beta >0\\[6pt]&=1-e^{-bsx},{\ }\beta =1\\\end{aligned}}

The moment generating function is given by:

{\begin{aligned}{\text{E}}(e^{-tx})={\begin{cases}\displaystyle \beta ^{s}{\frac {sb}{t+sb}}{\ }{_{2}{\text{F}}_{1}}(s+1,(t/b)+s;(t/b)+s+1;1-\beta ),&\beta \neq 1;\\\displaystyle {\frac {sb}{t+sb}},&\beta =1.\end{cases}}\end{aligned}}

where ${_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!$ is a Hypergeometric function.

The Gamma/Gompertz distribution is a flexible distribution that can be skewed to the right or to the left.

When β = 1, this reduces to an Exponential distribution with parameter sb.
The gamma distribution is a natural conjugate prior to a Gompertz likelihood with known, scale parameter $b\,\!.$ ^[1]
When the shape parameter $\eta \,\!$ of a Gompertz distribution varies according to a gamma distribution with shape parameter $\alpha \,\!$ and scale parameter $\beta \,\!$ (mean = $\alpha /\beta \,\!$ ), the distribution of $x$ is Gamma/Gompertz.^[1]

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Bemmaor, Albert C.; Glady, Nicolas (2011). "Implementing the Gamma/Gompertz/NBD Model in MATLAB" (PDF). Cergy-Pontoise: ESSEC Business School.^{[permanent dead link]}
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. S2CID 145157003.
Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1995). Continuous Univariate Distributions. Vol. 2 (2nd ed.). New York: John Wiley & Sons. pp. 25–26. ISBN 0-471-58494-0.
Manton, K. G.; Stallard, E.; Vaupel, J. W. (1986). "Alternative Models for the Heterogeneity of Mortality Risks Among the Aged". Journal of the American Statistical Association. 81 (395): 635–644. doi:10.1080/01621459.1986.10478316. PMID 12155405.