Probability density function edit

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

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

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

Cumulative distribution function edit

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

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

Moment generating function edit

The moment generating function is given by:

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

where ${\displaystyle {_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!}$  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 ${\displaystyle b\,\!.}$ ^[1]
• When the shape parameter ${\displaystyle \eta \,\!}$  of a Gompertz distribution varies according to a gamma distribution with shape parameter ${\displaystyle \alpha \,\!}$  and scale parameter ${\displaystyle \beta \,\!}$  (mean = ${\displaystyle \alpha /\beta \,\!}$ ), the distribution of ${\displaystyle x}$  is Gamma/Gompertz.^[1]

• Gompertz distribution
• Customer lifetime value

• 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. Archived from the original on 2015-06-26.
• 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.