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:
where is the scale parameter and are the shape parameters of the Gamma/Gompertz distribution.
Cumulative distribution function
The cumulative distribution function of the Gamma/Gompertz distribution is:
Moment generating function The moment generating function is given by:
where 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 When the shape parameter of a Gompertz distribution varies according to a gamma distribution with shape parameter and scale parameter (mean = ), the distribution of is Gamma/Gompertz.
See also Gompertz distribution Customer lifetime value
Notes
References
hu:Gompertz-eloszlás