Английская Википедия:Gamma/Gompertz distribution

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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:

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

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

Cumulative distribution function

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

<math>\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}</math>

Moment generating function

The moment generating function is given by:

<math>\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}</math> where <math> {_2\text{F}_1}(a,b;c;z) = \sum_{k=0}^\infty[(a)_k(b)_k/(c)_k]z^k/k!</math> 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 <math>b \,\!.</math>
• When the shape parameter <math>\eta\,\!</math> of a Gompertz distribution varies according to a gamma distribution with shape parameter <math>\alpha\,\!</math> and scale parameter <math>\beta\,\!</math> (mean = <math>\alpha/\beta\,\!</math>), the distribution of <math>x</math> is Gamma/Gompertz.

See also

• Gompertz distribution
• Customer lifetime value

References

• Шаблон:Cite journal
• Шаблон:Cite webШаблон:Dead link
• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite journal