Gamma/Gompertz distribution

Gamma/Gompertz distribution
	Probability density function; Note: b=0.4, β=3
	Cumulative distribution function
Parameters
Support
PDF
CDF	;
Mean	; ; ; ;
Median
Mode
Variance	; ; ; ; ; ; ; ; ; ;
MGF	; ; ; ;

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]

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.
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.

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Gamma/Gompertz distribution
Probability density function Note: b=0.4, β=3
Cumulative distribution function
Parameters	$b,s,\beta >0\,\!$
Support	$x\in [0,\infty )\!$
PDF	$bse^{bx}\beta ^{s}/\left(\beta -1+e^{bx}\right)^{s+1}{\text{where }}b,s,\beta >0$
CDF	$1-\beta ^{s}/\left(\beta -1+e^{bx}\right)^{s},x>0,b,s,\beta >0$ $1-e^{-bsx},\beta =1$
Mean	$=\left(1/b\right)\left(1/s\right){_{2}{\text{F}}_{1}}\left(s,1;s+1;\left(\beta -1\right)/\beta \right),$ $b,s>0,\beta \neq 1$ $=\left(1/b\right)\left[\beta /\left(\beta -1\right)\right]\ln \left(\beta \right),$ $b>0,s=1,\beta \neq 1$ $=1/\left(bs\right),\quad b,s>0,\beta =1$
Median	$\left(1/b\right)\ln\{\beta \left[\left(1/2\right)^{-1/s}-1\right]+1\}$
Mode	${\begin{aligned}x^{}&=(1/b)\ln \left[(1/s)(\beta -1)\right],\\&{\text{with }}0<{\text{F}}(x^{})<1-(\beta s)^{s}/\left[(\beta -1)(s+1)\right]^{s}<0.632121,\\&\beta >s+1\\&=0,\quad \beta \leq s+1\\\end{aligned}}$
Variance	$=2(1/b^{2})(1/s^{2})\beta ^{s}{_{3}{\text{F}}_{2}}(s,s,s;s+1,s+1;1-\beta )$ $-{\text{E}}^{2}(\tau \|b,s,\beta ),\quad \beta \neq 1$ $=(1/b^{2})(1/s^{2}),\quad \beta =1$ ${\text{with}}$ ${_{3}{\text{F}}_{2}}(a,b,c;d,e;z)=\sum _{k=0}^{\infty }\{(a)_{k}(b)_{k}(c)_{k}/[(d)_{k}(e)_{k}]\}z^{k}/k!$ ${\text{and}}$ $(a)_{k}=\Gamma (a+k)/\Gamma (a)$
MGF	${\text{E}}(e^{-tx})$ $=\beta ^{s}[sb/(t+sb)]{_{2}{\text{F}}_{1}}(s+1,(t/b)+s;(t/b)+s+1;1-\beta ),$ $\quad \beta \neq 1$ $=sb/(t+sb),\quad \beta =1$ ${\text{with }}{_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!$