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Shifted Gompertz

Probability density function
Cumulative distribution function
Parameters	$b>0$ scale (real) $\eta >0$ shape (real)
Support	$x\in \mathbb {R} ^{+}$
PDF	$be^{-bx}e^{-\eta e^{-bx}}\left[1+\eta \left(1-e^{-bx}\right)\right]$
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 ${\begin{aligned}\mathrm {E} [\ln(X)]=&[1{+}1/\eta ]\!\!\int _{0}^{\eta }\!\!\!\!e^{-X}[\ln(X)]dX\\&-1/\eta \!\!\int _{0}^{\eta }\!\!\!\!Xe^{-X}[\ln(X)]dX\end{aligned}}$
Mode	$0\,$ for $\eta \leq 0.5\,$, $(-1/b)\ln(x^{\star })\,$ for $\eta >0.5\,$where$x^{\star }=[3+\eta -(\eta ^{2}+2\eta +5)^{1/2}]/(2\eta )$
Variance	$(1/b^{2})(\mathrm {E} [\ln(X)^{2}]-(\mathrm {E} [\ln(X)])^{2})\,$ where $X=\eta e^{-bx}\,$ and ${\begin{aligned}\mathrm {E} [\ln(X)^{2}]=&[1{+}1/\eta ]\!\!\int _{0}^{\eta }\!\!\!\!e^{-X}[\ln(X)]^{2}dX\\&-1/\eta \!\!\int _{0}^{\eta }\!\!\!\!Xe^{-X}[\ln(X)]^{2}dX\end{aligned}}$

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:

$f(x;b,\eta )=be^{-bx}e^{-\eta e^{-bx}}\left[1+\eta \left(1-e^{-bx}\right)\right]\mathrm {for} \ x>0\,\!$

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

Cumulative distribution function

The cumulative distribution function of the shifted Gompertz distribution is:

$F(x;b,\eta )=\left(1-e^{-bx}\right)e^{-\eta e^{-bx}}\mathrm {for} \ x>0\,\!$

Properties

The shifted Gompertz distribution is right-skewed for all values of $\eta$.

Shapes

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 $(-1/b)\ln(x^{\star })\,,0<x^{\star }<1$ where $x^{\star }\,$ is the smallest root of $\eta ^{2}x^{2}-\eta (3+\eta )x+\eta +1=0\,$ which is $x^{\star }=[3+\eta -(\eta ^{2}+2\eta +5)^{1/2}]/(2\eta )$

Related Distributions

If $\eta$ varies according to a gamma distribution with shape parameter $\alpha$ and scale parameter $\beta$ (mean = $\alpha \beta$), the cumulative distribution function is Gamma/Shifted Gompertz.

References

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 (ed.), Research Traditions in Marketing, Boston: Kluwer Academic Publishers, pp. 201–223. Van Den Bulte, Christophe (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. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) Chandrasekaran, Deepa; Tellis, Gerard J. (2007), "A Critical Review of Marketing Research on Diffusion of New Products", in Naresh K. Malhotra (ed.), Review of Marketing Research, vol. 3, Armonk: M.E. Sharpe.