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.
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, the distribution of is Gamma/Gompertz.