Gompertz distribution

In probability and statistics, the Gompertz distribution is a continuous probability distribution, named after Benjamin Gompertz. The Gompertz distribution is often applied to describe the distribution of adult lifespans by demographers and actuaries. Related fields of science such as biology and gerontology also considered the Gompertz distribution for the analysis of survival. More recently, computer scientists have also started to model the failure rates of computer code by the Gompertz distribution. In marketing, it has been used as an individual-level simulation for customer lifetime value modeling. In network theory, particularly the Erdős–Rényi model, the walk length of a random self-avoiding walk (SAW) is distributed according to the Gompertz distribution.

Specification

Probability density function The probability density function of the Gompertz distribution is:

f ( x ; η , b ) = b η exp ⁡ ( η + b x − η e b x ) for x ≥ 0 , {\displaystyle f\left(x;\eta ,b\right)=b\eta \exp \left(\eta +bx-\eta e^{bx}\right){\text{for }}x\geq 0,\,}

where b > 0 {\displaystyle b>0\,\!} is the scale parameter and η > 0 {\displaystyle \eta >0\,\!} is the shape parameter of the Gompertz distribution. In the actuarial and biological sciences and in demography, the Gompertz distribution is parametrized slightly differently (Gompertz–Makeham law of mortality).

Cumulative distribution function The cumulative distribution function of the Gompertz distribution is:

F ( x ; η , b ) = 1 − exp ⁡ ( − η ( e b x − 1 ) ) , {\displaystyle F\left(x;\eta ,b\right)=1-\exp \left(-\eta \left(e^{bx}-1\right)\right),}

where η , b > 0 , {\displaystyle \eta ,b>0,} and x ≥ 0 . {\displaystyle x\geq 0\,.}

Moment generating function The moment generating function is:

E ( e − t X ) = η e η E t / b ( η ) {\displaystyle {\text{E}}\left(e^{-tX}\right)=\eta e^{\eta }{\text{E}}_{t/b}\left(\eta \right)}

where

E t / b ( η ) = ∫ 1 ∞ e − η v v − t / b d v , t > 0. {\displaystyle {\text{E}}_{t/b}\left(\eta \right)=\int _{1}^{\infty }e^{-\eta v}v^{-t/b}dv,\ t>0.}

Properties The Gompertz distribution is a flexible distribution that can be skewed to the right and to the left. Its hazard function h ( x ) = η b e b x {\displaystyle h(x)=\eta be^{bx}} is a convex function of F ( x ; η , b ) {\displaystyle F\left(x;\eta ,b\right)} . The model can be fitted into the innovation-imitation paradigm with

p = η b {\displaystyle p=\eta b} as the coefficient of innovation and b {\displaystyle b} as the coefficient of imitation. When t {\displaystyle t} becomes large, z ( t ) {\displaystyle z(t)} approaches ∞ {\displaystyle \infty } . The model can also belong to the propensity-to-adopt paradigm with

η {\displaystyle \eta } as the propensity to adopt and b {\displaystyle b} as the overall appeal of the new offering.

Shapes The Gompertz density function can take on different shapes depending on the values of the shape parameter η {\displaystyle \eta \,\!} :

When η ≥ 1 , {\displaystyle \eta \geq 1,\,} the probability density function has its mode at 0. When 0 < η < 1 , {\displaystyle 0<\eta <1,\,} the probability density function has its mode at

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