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 and to the left.

Related distributions

See also

Notes

  1. ↑ 1.0 1.1 Bemmaor, A.C.; Glady, N. (2012)

References