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)
η > 0 shape (real)
Support x \in \mathbb{R}^+
Probability density function (pdf) b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]
Cumulative distribution function (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\, for \eta \leq 0.5\,, (-1/b)\ln(z^\star)\, for \eta > 0.5\,wherez^\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
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

The shifted Gompertz distribution is the distribution of the largest order statistic of two independent random variables which are distributed exponential and Gompertz with parameters b and b and η respectively. It has been used as a model of adoption of innovation.

Contents

[edit] Specification

[edit] 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] \mathrm{for}\ x > 0 \,\!

where b > 0 is the scale parameter and η > 0 is the shape parameter of the shifted Gompertz distribution.

[edit] 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}} \mathrm{for}\ x > 0. \,\!

[edit] Properties

The shifted Gompertz distribution is right-skewed for all values of η.

[edit] Shapes

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

  • \eta \leq 0.5\, the probability density function has mode 0.
  • \eta > 0.5\, the probability density function has the mode at (-1/b)\ln(z^\star)\,, 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)

[edit] Related distributions

If η varies according to a gamma distribution with shape parameter α and scale parameter β (mean = αβ), the cumulative distribution function is Gamma/Shifted Gompertz.

[edit] See also

[edit] References

  • Bemmaor, Albert C. (1994), "Modeling the Diffusion of New Durable Goods: Word-of-Mouth Effect Versus Consumer Heterogeneity", written at Boston, in G. Laurent, G.L. Lilien & B. Pras, Research Traditions in Marketing, Kluwer Academic Publishers, 201-223.
  • Van Den Bulte, Christophe; Stefan Stremersch (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. 
  • Chandrasekaran, Deepa & Gerard J. Tellis (2007), "A Critical Review of Marketing Research on Diffusion of New Products", written at Armonk, in Naresh K. Malhotra, Review of Marketing Research, vol. 3, M.E. Sharpe.
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Continuous univariate supported on a semi-infinite interval