The shifted Gompertz distribution is the distribution of the larger of two independent random variables one of which has an exponential distribution with parameter b {\displaystyle b} and the other has a Gumbel distribution with parameters η {\displaystyle \eta } and b {\displaystyle b} . In its original formulation the distribution was expressed referring to the Gompertz distribution instead of the Gumbel distribution but, since the Gompertz distribution is a reverted Gumbel distribution, the labelling can be considered as accurate. It has been used as a model of adoption of innovations. It was proposed by Bemmaor (1994). Some of its statistical properties have been studied further by Jiménez and Jodrá (2009) and Jiménez Torres (2014). It has been used to predict the growth and decline of social networks and on-line services and shown to be superior to the Bass model and Weibull distribution (Bauckhage and Kersting 2014).
Specification
Probability density function The probability density function of the shifted Gompertz distribution is:
f ( x ; b , η ) = b e − b x e − η e − b x [ 1 + η ( 1 − e − b x ) ] for x ≥ 0. {\displaystyle f(x;b,\eta )=be^{-bx}e^{-\eta e^{-bx}}\left[1+\eta \left(1-e^{-bx}\right)\right]{\text{ for }}x\geq 0.\,}
where b ≥ 0 {\displaystyle b\geq 0} is a scale parameter and η ≥ 0 {\displaystyle \eta \geq 0} is a shape parameter. In the context of diffusion of innovations, b {\displaystyle b} can be interpreted as the overall appeal of the innovation and η {\displaystyle \eta } is the propensity to adopt in the propensity-to-adopt paradigm. The larger b {\displaystyle b} is, the stronger the appeal and the larger η {\displaystyle \eta } is, the smaller the propensity to adopt. The distribution can be reparametrized according to the external versus internal influence paradigm with p = f ( 0 ; b , η ) = b e − η {\displaystyle p=f(0;b,\eta )=be^{-\eta }} as the coefficient of external influence and q = b − p {\displaystyle q=b-p} as the coefficient of internal influence. Hence:
f ( x ; p , q ) = ( p + q ) e − ( p + q ) x e − ln ( 1 + q / p ) e − ( p + q ) x [ 1 + ln ( 1 + q / p ) ( 1 − e − ( p + q ) x ) ] for x ≥ 0 , p , q ≥ 0. {\displaystyle f(x;p,q)=(p+q)e^{-(p+q)x}e^{-\ln(1+q/p)e^{-(p+q)x}}\left[1+\ln(1+q/p)\left(1-e^{-(p+q)x}\right)\right]{\text{ for }}x\geq 0,p,q\geq 0.\,}
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