Gamma/Gompertz distribution

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	$=\left(1/b\right)\left(1/s\right){_2\text{F}_1}\left(s,1;s+1;\left(\beta-1\right)/\beta\right),$            $b,s>0, \beta\ne1$ $=\left(1/b\right)\left[\beta/\left(\beta-1\right)\right]\ln\left(\beta\right),$            $b>0,s=1,\beta\ne1$ $=1/\left(bs\right),\quad b,s>0,\beta=1$
Median	$\left(1/b\right)\ln\{\beta\left[\left(1/2\right)^{-1/s}-1\right]+1\}$
Mode	$\begin{align}x^*& = (1/b)\ln\left[(1/s)(\beta-1)\right], \\&\text{with } 0<\text{F}(x^*)<1-(\beta s)^s/\left[(\beta-1)(s+1)\right]^s<0.632121,\\& \beta > s+1\\& = 0, \quad \beta\le s+1\\\end{align}$
Variance	$=2(1/b^{2})(1/s^{2})\beta^{s} {_3\text{F}_2}(s,s,s;s+1,s+1;1-\beta)$            $- \text{E}^{2}(\tau|b,s,\beta), \quad \beta \ne 1$ $=(1/b^{2})(1/s^{2}), \quad \beta = 1$ $\text{with}$ ${_3\text{F}_2}(a,b,c;d,e;z) = \sum_{k=0}^\infty\{(a)_k(b)_k(c)_k/[(d)_k(e)_k]\}z^k/k!$ $\text{and}$ $(a)_k=\Gamma(a+k)/\Gamma(a)$
MGF	$\text{E}(e^{-tx})$ $=\beta^{s}[sb/(t+sb)]{_2\text{F}_1}(s+1,(t/b)+s;(t/b)+s+1;1-\beta),$ $\quad \beta \ne 1$ $= sb/(t+sb), \quad \beta =1$ $\text{with }{_2\text{F}_1}(a,b;c;z) = \sum_{k=0}^\infty[(a)_k(b)_k/(c)_k]z^k/k!$

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:

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

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

Cumulative distribution function

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

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

Moment generating function

The moment generating function is given by:

$\begin{align} \text{E}(e^{-tx})= \begin{cases}\displaystyle \beta^s \frac{sb}{t+sb}{\ } {_2\text{F}_1}(s+1,(t/b)+s;(t/b)+s+1;1-\beta), & \beta \ne 1; \\ \displaystyle \frac{sb}{t+sb},& \beta =1. \end{cases} \end{align}$

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.

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 $b \,\!.$^[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]

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.
• 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: 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: 635–644. doi:10.1080/01621459.1986.10478316.