Shifted Gompertz distribution information

Shifted Gompertz
	Probability density function
	Cumulative distribution function
Parameters	scale (real); shape (real)
Support
PDF
CDF
Mean	where and
Mode	; ;
Variance	where and

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^[1] (1994). Some of its statistical properties have been studied further by Jiménez and Jodrá ^[2](2009) and Jiménez Torres ^[3](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^[4] 2014).

^ Bemmaor, Albert C. (1994). "Modeling the Diffusion of New Durable Goods: Word-of-Mouth Effect Versus Consumer Heterogeneity". In G. Laurent, G.L. Lilien & B. Pras (ed.). Research Traditions in Marketing. Boston: Kluwer Academic Publishers. pp. 201–223. ISBN 978-0-7923-9388-7.
^ Jiménez, Fernando; Jodrá, Pedro (2009). "A Note on the Moments and Computer Generation of the Shifted Gompertz Distribution". Communications in Statistics - Theory and Methods. 38 (1): 78–89. doi:10.1080/03610920802155502. S2CID 116954940.
^ Jiménez Torres, Fernando (2014). "Estimation of the Parameters of the Shifted Gompertz Distribution, Using Least Squares, Maximum Likelihood and Moments Methods". Journal of Computational and Applied Mathematics. 255 (1): 867–877. doi:10.1016/j.cam.2013.07.004.
^ Bauckhage, Christian; Kersting, Kristian (2014). "Strong Regularities in Growth and Decline of Popularity of Social Media Services". arXiv:1406.6529 [math-ph].

[1] Bemmaor, Albert C. (1994). "Modeling the Diffusion of New Durable Goods: Word-of-Mouth Effect Versus Consumer Heterogeneity". In G. Laurent, G.L. Lilien & B. Pras (ed.). Research Traditions in Marketing. Boston: Kluwer Academic Publishers. pp. 201–223. ISBN 978-0-7923-9388-7.

[2] Jiménez, Fernando; Jodrá, Pedro (2009). "A Note on the Moments and Computer Generation of the Shifted Gompertz Distribution". Communications in Statistics - Theory and Methods. 38 (1): 78–89. doi:10.1080/03610920802155502. S2CID 116954940.

[3] Jiménez Torres, Fernando (2014). "Estimation of the Parameters of the Shifted Gompertz Distribution, Using Least Squares, Maximum Likelihood and Moments Methods". Journal of Computational and Applied Mathematics. 255 (1): 867–877. doi:10.1016/j.cam.2013.07.004.

[4] Bauckhage, Christian; Kersting, Kristian (2014). "Strong Regularities in Growth and Decline of Popularity of Social Media Services". arXiv:1406.6529 [math-ph].

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Shifted Gompertz
Probability density function
Cumulative distribution function
Parameters	$b\geq 0$ scale (real) $\eta \geq 0$ shape (real)
Support	$x\in [0,\infty )\!$
PDF	$be^{-bx}e^{-\eta e^{-bx}}\left[1+\eta \left(1-e^{-bx}\right)\right]$
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 ${\begin{aligned}\mathrm {E} [\ln(X)]=&[1{+}1/\eta ]\!\!\int _{0}^{\eta }\!\!\!\!e^{-X}[\ln(X)]dX\\&-1/\eta \!\!\int _{0}^{\eta }\!\!\!\!Xe^{-X}[\ln(X)]dX\end{aligned}}$
Mode	$0{\text{ for }}0<\eta \leq 0.5$ $(-1/b)\ln(z^{\star }){\text{, for }}\eta >0.5$ ${\text{ where }}z^{\star }=[3+\eta -(\eta ^{2}+2\eta +5)^{1/2}]/(2\eta )$
Variance	$(1/b^{2})(\mathrm {E} \{[\ln(X)]^{2}\}-(\mathrm {E} [\ln(X)])^{2})\,$ where $X=\eta e^{-bx}\,$ and ${\begin{aligned}\mathrm {E} \{[\ln(X)]^{2}\}=&[1{+}1/\eta ]\!\!\int _{0}^{\eta }\!\!\!\!e^{-X}[\ln(X)]^{2}dX\\&-1/\eta \!\!\int _{0}^{\eta }\!\!\!\!Xe^{-X}[\ln(X)]^{2}dX\end{aligned}}$