Shifted Gompertz distribution

Shifted Gompertz distribution

Probability distribution
name = Shifted Gompertz
type =density
pdf_

cdf_

parameters =b>0 scale (real) eta>0 shape (real)
support =x in mathbb{R}^+
pdf =b e^{-bx} e^{-eta e^{-bx
left [1 + etaleft(1 - e^{-bx} ight) ight]
cdf =left(1 - e^{-bx} ight)e^{-eta e^{-bx
mean =(-1/b){mathrm{E} [ln(X)] - ln(eta)}, where X = eta e^{-bx}, and egin{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 + 2eta + 5)^{1/2}] /(2eta)
variance =(1/b^2)(mathrm{E}{ [ln(X)] ^2} - (mathrm{E} [ln(X)] )^2), where X = eta e^{-bx}, and egin{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 largest order statistic of two independent random variables which are distributed exponential and Gompertz with parameters b and b and eta respectively. It has been used as a model of adoption of innovation.

Specification

Probability density function

The probability density function of the shifted Gompertz distribution is:

: f(x;b,eta) = b e^{-bx} e^{-eta e^{-bxleft [1 + etaleft(1 - e^{-bx} ight) ight] mathrm{for} x > 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} ight)e^{-eta e^{-bx mathrm{for} x > 0. ,!

Properties

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

hapes

The shifted Gompertz density function can take on different shapes depending on the values of the shape parameter eta:
* 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 + 2eta + 5)^{1/2}] /(2eta)

Related distributions

If eta varies according to a gamma distribution with shape parameter alpha and scale parameter eta (mean = alphaeta), the cumulative distribution function is Gamma/Shifted Gompertz (G/SG). When alpha is equal to one, the G/SG reduces to the Bass model.

See also

*Gumbel distribution
*Extreme value distribution
*Mixture model
*Bass model

References

*Harvard reference | Surname=Bemmaor | Given=Albert C. | Year= 1994 |pages=201-223| Chapter=Modeling the Diffusion of New Durable Goods: Word-of-Mouth Effect Versus Consumer Heterogeneity | Editor=G. Laurent, G.L. Lilien & B. Pras | Title=Research Traditions in Marketing | Publisher=Kluwer Academic Publishers | Place=Boston.

*cite journal
last = Van Den Bulte
first = Christophe
title = Social Contagion and Income Heterogeneity in New Product Diffusion: A Meta-Analytic Test
coauthors = Stefan Stremersch
journal = Marketing Science
volume = 23
issue = 4
pages = 530–544
year = 2004
doi = 10.1287/mksc.1040.0054

*Harvard reference | Surname1=Chandrasekaran| Given1=Deepa | Surname2=Tellis| Given2=Gerard J. |Year= 2007 | Volume=3| Chapter=A Critical Review of Marketing Research on Diffusion of New Products | Editor=Naresh K. Malhotra | Title=Review of Marketing Research | Publisher=M.E. Sharpe | Place=Armonk.


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