[3] 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).

[4] The probability density function of the shifted Gompertz distribution is: where

In the context of diffusion of innovations,

can be interpreted as the overall appeal of the innovation and

is the propensity to adopt in the propensity-to-adopt paradigm.

is, the stronger the appeal and the larger

is, the smaller the propensity to adopt.

The distribution can be reparametrized according to the external versus internal influence paradigm with

, the proportion of adopters is nil: the innovation is a complete failure.

The shape parameter of the probability density function is equal to

Similar to the Bass model, the hazard rate

See Bemmaor and Zheng [5] for further analysis.

The cumulative distribution function of the shifted Gompertz distribution is: Equivalently, The shifted Gompertz distribution is right-skewed for all values of

The hazard rate is a concave function of

In the context of the diffusion of innovations, the effect of word of mouth (i.e., the previous adopters) on the likelihood to adopt decreases as the proportion of adopters increases.

(For comparison, in the Bass model, the effect remains the same over time).

captures the growth of the hazard rate when

The shifted Gompertz density function can take on different shapes depending on the values of the shape parameter

varies according to a gamma distribution with shape parameter

is equal to one, the G/SG reduces to the Bass model (Bemmaor 1994).

The three-parameter G/SG has been applied by Dover, Goldenberg and Shapira (2009)[6] and Van den Bulte and Stremersch (2004)[7] among others in the context of the diffusion of innovations.

The model is discussed in Chandrasekaran and Tellis (2007).

[8] Similar to the shifted Gompertz distribution, the G/SG can either be represented according to the propensity-to-adopt paradigm or according to the innovation-imitation paradigm.

modifies the curvature of the hazard rate as expressed as a function of

is less than one and larger or equal to 0.5, linear when

Here are some special cases of the G/SG distribution in the case of homogeneity (across the population) with respect to the likelihood to adopt at a given time: with: One can compare the parameters

In all the cases, the hazard rate is either constant or a monotonically increasing function of

(positive word of mouth).

As the diffusion curve is all the more skewed as

to decrease as the level of right-skew increases.