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Gamma/Gompertz distribution

Probability distribution

Probability distribution

Note: b=0.4, β=3 1-e^{-bsx}, \beta=1 b,s0, \beta\ne1 =\left(1/b\right)\left[\beta/\left(\beta-1\right)\right]\ln\left(\beta\right), b0,s=1,\beta\ne1 =1/\left(bs\right),\quad b,s0,\beta=1 - \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) =\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}, {\ }x0, {\ } b,s,\beta0 \[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.

Notes

References

  • {{Cite journal | last=Gompertz | first=B. | year= 1825 |pages=513–583| title=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 | journal =Philosophical Transactions of the Royal Society of London| volume = 115 |jstor=107756 | doi=10.1098/rstl.1825.0026| s2cid=145157003 | url=https://zenodo.org/record/1432356 | doi-access=free }}

hu:Gompertz-eloszlás

References

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