Gamma/Gompertz distribution

Gamma/Gompertz distribution

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

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

Probability density function

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

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

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.

Properties

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

Related distributions

• 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]

See also

• Gompertz distribution
• Customer lifetime value

Notes

1. Bemmaor, A.C.; Glady, N. (2012)

References

• 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. 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.