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
median =
mode = $0,$ for $eta\; leq\; 0.5,$, $(-1/b)ln(z^star),$ for $eta\; >\; 0.5,$where$z^star\; =\; [3\; +\; eta\; -\; (eta^2\; +\; 2eta\; +\; 5)^\{1/2\}]\; /(2eta)$
variance =$(1/b^2)(mathrm\{E\}\{\; [ln(X)]\; ^2\}\; -\; (mathrm\{E\}\; [ln(X)]\; )^2),$ where $X\; =\; eta\; e^\{-bx\},$ and
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 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 (mean = ), 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.