Shifted Gompertz distribution

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Shifted Gompertz
Probability density function
Probability density plots of shifted Gompertz distributions
Cumulative distribution function
Cumulative distribution plots of shifted Gompertz distributions
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[edit]

Probability density function[edit]

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[edit]

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[edit]

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

Shapes[edit]

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[edit]

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[edit]

References[edit]

  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.