Shifted Gompertz distribution

Shifted Gompertz

Probability density function
Cumulative distribution function
Parameters	$b>0$ scale (real) $\eta>0$ 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 $\begin{align}\mathrm{E}[\ln(X)] =& [1 {+} 1 / \eta]\!\!\int_0^\eta \!\!\!\! e^{-X}[\ln(X)]dX\\ &- 1/\eta\!\! \int_0^\eta \!\!\!\! X e^{-X}[\ln(X)] dX \end{align}$
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 $\begin{align}\mathrm{E}\{[\ln(X)]^2\} =& [1 {+} 1 / \eta]\!\!\int_0^\eta \!\!\!\! e^{-X}[\ln(X)]^2 dX\\ &- 1/\eta \!\!\int_0^\eta \!\!\!\! X e^{-X}[\ln(X)]^2 dX \end{align}$

The shifted Gompertz distribution is the distribution of the largest 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).

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 (see the work by Christian Bauckhage and co-authors).

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 $b > 0$ is the scale parameter and $\eta > 0$ is the shape parameter of the shifted Gompertz distribution.

Cumulative distribution function

The cumulative distribution function of the shifted Gompertz distribution is:

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

Properties

The shifted Gompertz distribution is right-skewed for all values of $\eta$. It is more flexible than the Gumbel distribution.

Shapes

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

• $0 < \eta \leq 0.5\,$ the probability density function has its mode at 0.
• $\eta > 0.5\,$ the probability density function has its mode at

$\text{mode}=-\frac{\ln(z^\star)}{b}\, \qquad 0 < z^\star < 1$

where $z^\star\,$ is the smallest root of

$\eta^2z^2 - \eta(3 + \eta)z + \eta + 1 = 0\,,$

which is

$z^\star = [3 + \eta - (\eta^2 + 2\eta + 5)^{1/2}]/(2\eta).$

Related distributions

If $\eta$ 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 G/SG has been applied by Dover, Goldenberg and Shapira ^[3](2009) and Van den Bulte and Stremersch ^[4](2004) among others in the context of the diffusion of innovations. The model is discussed in Chandrasekaran and Tellis ^[5](2007).

See also

• Gumbel distribution
• Generalized extreme value distribution
• Mixture model
• Bass model
• Gompertz distribution

References

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. Research Traditions in Marketing. Boston: Kluwer Academic Publishers. pp. 201–223. ISBN 0-7923-9388-0.
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.
3. Dover, Yaniv; Goldenberg, Jacob; Shapira, Daniel (2012). "Network Traces on Penetration: Uncovering Degree Distribution From Adoption Data". Marketing Science. doi:10.1287/mksc.1120.0711.
4. Van den Bulte, Christophe; Stremersch, Stefan (2004). "Social Contagion and Income Heterogeneity in New Product Diffusion: A Meta-Analytic Test". Marketing Science 23 (4): 530–544. doi:10.1287/mksc.1040.0054.
5. Chandrasekaran, Deepa; Tellis, Gerard J. (2007). "A Critical Review of Marketing Research on Diffusion of New Products". In Naresh K. Malhotra. Review of Marketing Research 3. Armonk: M.E. Sharpe. pp. 39–80. ISBN 978-0-7656-1306-6.