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Shifted Gompertz distribution

Probability distribution

name = Shifted Gompertz| type =density| pdf_image =[[File:Shiftedgompertz distribution PDF.svg|325px|Probability density plots of shifted Gompertz distributions]]| cdf_image =[[File:Shiftedgompertz distribution CDF.svg|325px|Cumulative distribution plots of shifted Gompertz distributions]]| parameters =b \geq 0 scale (real) \eta\geq 0 shape (real)| support =x \in [0, \infty)!| pdf =b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]| cdf =\left(1 - e^{-bx}\right)e^{-\eta e^{-bx}}| mean =(-1/b){\mathrm{E}[\ln(X)] - \ln(\eta)}, where X = \eta e^{-bx}, and \begin{align}\mathrm{E}[\ln(X)] =& [1 {+} 1 / \eta]!!\int_0^\eta !!!! e^{-X}[\ln(X)]dX\ &- 1/\eta!! \int_0^\eta !!!! X e^{-X}[\ln(X)] dX \end{align}| median =| mode = 0 \text{ for }0 (-1/b)\ln(z^\star)\text{, for } \eta 0.5 \text{ where }z^\star = [3 + \eta - (\eta^2 + 2\eta + 5)^{1/2}]/(2\eta)| variance =(1/b^2)(\mathrm{E}{[\ln(X)]^2} - (\mathrm{E}[\ln(X)])^2), where X = \eta e^{-bx}, and \begin{align}\mathrm{E}{[\ln(X)]^2} =& [1 {+} 1 / \eta]!!\int_0^\eta !!!! e^{-X}[\ln(X)]^2 dX\ &- 1/\eta !!\int_0^\eta !!!! X e^{-X}[\ln(X)]^2 dX \end{align}| skewness =| kurtosis =| entropy =| mgf =| char =| 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 and the other has a Gumbel distribution with parameters \eta and 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){{Cite journal and Jiménez Torres (2014).{{Cite journal

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,\eta) = b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right] \text{ for }x \geq 0. ,

where b \geq 0 is a scale parameter and \eta \geq 0 is a shape parameter. In the context of diffusion of innovations, b can be interpreted as the overall appeal of the innovation and \eta is the propensity to adopt in the propensity-to-adopt paradigm. The larger b is, the stronger the appeal and the larger \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,\eta) = be^{-\eta } as the coefficient of external influence and 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}}\left[1 + \ln(1 + q/p)\left(1 - e^{-(p + q)x}\right)\right] \text{ for }x \geq 0, p, q \geq 0. , : = (p + q) e^{-(p + q)x} {(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. ,

When q = 0 , the shifted Gompertz distribution reduces to an exponential distribution. When p = 0, the proportion of adopters is nil: the innovation is a complete failure. The shape parameter of the probability density function is equal to q/p . Similar to the Bass model, the hazard rate z(x;p,q) is equal to p when x is equal to 0 ; it approaches p + q as x gets close to \infty. See Bemmaor and Zheng {{Cite journal

Cumulative distribution function

The cumulative distribution function of the shifted Gompertz distribution is:

: F(x;b,\eta) = \left(1 - e^{-bx}\right)e^{-\eta e^{-bx}} \text{ for }x \geq 0. ,

Equivalently,

: F(x;p, q) = \left(1 - e^{-(p + q)x}\right)e^{-\ln(1 + q/p)e^{-(p+q)x}} \text{ for }x \geq 0. , : = \left(1 - e^{-(p + q)x}\right){(1 + q/p)^{-e^{-(p+q)x}}} \text{ for }x \geq 0. ,

Properties

The shifted Gompertz distribution is right-skewed for all values of \eta. It is more flexible than the Gumbel distribution. The hazard rate is a concave function of F(x;b,\eta) which increases from be^{-\eta} to b : its curvature is all the steeper as \eta is large. 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). The parameter q = b(1-e^{-\eta}) captures the growth of the hazard rate when x varies from 0 to \infty.

Shapes

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

0 the probability density function has its mode at 0.
\eta 0.5, the probability density function has its mode at ::\text{mode}=-\frac{\ln(z^\star)}{b}, \qquad 0 :where z^\star, is the smallest root of ::\eta^2z^2 - \eta(3 + \eta)z + \eta + 1 = 0,, :which is

## Related distributions When \eta varies according to a gamma distribution with shape parameter \alpha and scale parameter \beta (mean = \alpha\beta), the distribution of x is Gamma/Shifted Gompertz (G/SG). When \alpha 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){{Cite journal and Van den Bulte and Stremersch (2004){{Cite journal among others in the context of the diffusion of innovations. The model is discussed in Chandrasekaran and Tellis (2007). 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. In the latter case, it includes three parameters: p, q and \alpha with p = f(0;b,\beta, \alpha) = b/(1+\beta)^{\alpha } and q = b - p . The parameter \alpha modifies the curvature of the hazard rate as expressed as a function of F(x;p,q, \alpha): when \alpha is less than 0.5, it decreases to a minimum prior to increasing at an increasing rate as F(x;p,q, \alpha increases, it is convex when \alpha is less than one and larger or equal to 0.5, linear when \alpha is equal to one, and concave when \alpha is larger than one. 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: F(x;p,q, \alpha = 0) = Exponential(p + q) F(x;p,q, \alpha = 1/2) = Left-skewed two-parameter distribution(p,q) F(x;p,q, \alpha = 1) = Bass model(p,q) F(x;p,q, \alpha = \infty) = Shifted Gompertz(p,q) with: F(x;p, q,\alpha = 1/2) = \left(1 - e^{-(p + q)x}\right)/{(1 + (q/p)(2+q/p)e^{-(p+q)x})^{1/2}} \text{ for }x \geq 0,p, q \geq 0. \, One can compare the parameters p and q across the values of \alpha as they capture the same notions. In all the cases, the hazard rate is either constant or a monotonically increasing function of F(x;p,q, \alpha) (positive word of mouth). As the diffusion curve is all the more skewed as \alpha becomes large, we expect q to decrease as the level of right-skew increases. ## References ## References 1. Bemmaor. (1994). "Research Traditions in Marketing". *Kluwer Academic Publishers*. 2. (2014). "Strong Regularities in Growth and Decline of Popularity of Social Media Services". ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Shifted_Gompertz_distribution) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Shifted_Gompertz_distribution?action=history). ::

continuous-distributions

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