Gamma/Gompertz distribution

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.

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 &beta; = 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 $$b \,\!.$$
 * 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.