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

The probability density function of the Gamma/Gompertz distribution is:

f (x; b, s, β) = \frac{b s e^{b x} β^{s}}{{(β - 1 + e^{b x})}^{s + 1}}

where $b > 0$ is the scale parameter and $β, s > 0$ are the shape parameters of the Gamma/Gompertz distribution.

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

\begin{matrix} F (x; b, s, β) & = 1 - \frac{β^{s}}{{(β - 1 + e^{b x})}^{s}}, x > 0, b, s, β > 0 \\ = 1 - e^{- b s x}, β = 1 \end{matrix}

The moment generating function is given by:

\begin{matrix} E (e^{- t x}) = {\begin{matrix} β^{s} \frac{s b}{t + s b}_{2} F_{1} (s + 1, (t / b) + s; (t / b) + s + 1; 1 - β), & β \neq 1; \\ \frac{s b}{t + s b}, & β = 1 . \end{matrix} \end{matrix}

where $_{2} F_{1} (a, b; c; z) = \sum_{k = 0}^{\infty} [(a)_{k} (b)_{k} / (c)_{k}] z^{k} / k!$ is a Hypergeometric function.

The Gamma/Gompertz distribution is a flexible distribution that can be skewed to the right or to the left.

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 $b .$ ^[1]
When the shape parameter $η$ of a Gompertz distribution varies according to a gamma distribution with shape parameter $α$ and scale parameter $β$ (mean = $α / β$ ), the distribution of $x$ is Gamma/Gompertz.^[1]

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