Probability density function

The probability density function of the shifted Gompertz distribution is:

${\displaystyle f(x;b,\eta )=be^{-bx}e^{-\eta e^{-bx}}\left[1+\eta \left(1-e^{-bx}\right)\right]{\text{ for }}x\geq 0.\,}$

where ${\displaystyle b\geq 0}$ is a scale parameter and ${\displaystyle \eta \geq 0}$ is a shape parameter. In the context of diffusion of innovations, ${\displaystyle b}$ can be interpreted as the overall appeal of the innovation and ${\displaystyle \eta }$ is the propensity to adopt in the propensity-to-adopt paradigm. The larger ${\displaystyle b}$ is, the stronger the appeal and the larger ${\displaystyle \eta }$ is, the smaller the propensity to adopt.

The distribution can be reparametrized according to the external versus internal influence paradigm with ${\displaystyle p=f(0;b,\eta )=be^{-\eta }}$ as the coefficient of external influence and ${\displaystyle q=b-p}$ as the coefficient of internal influence. Hence:

${\displaystyle f(x;p,q)=(p+q)e^{-(p+q)x}e^{-\ln(1+q/p)e^{-(p+q)x}}\left[1+\ln(1+q/p)\left(1-e^{-(p+q)x}\right)\right]{\text{ for }}x\geq 0,p,q\geq 0.\,}$

${\displaystyle =(p+q)e^{-(p+q)x}{(1+q/p)^{-e^{-(p+q)x}}}\left[1+\ln(1+q/p)\left(1-e^{-(p+q)x}\right)\right]{\text{ for }}x\geq 0,p,q\geq 0.\,}$

When ${\displaystyle q=0}$, the shifted Gompertz distribution reduces to an exponential distribution. When ${\displaystyle p=0}$, the proportion of adopters is nil: the innovation is a complete failure. The shape parameter of the probability density function is equal to ${\displaystyle q/p}$. Similar to the Bass model, the hazard rate ${\displaystyle z(x;p,q)}$ is equal to ${\displaystyle p}$ when ${\displaystyle x}$ is equal to ${\displaystyle 0}$; it approaches ${\displaystyle p+q}$ as ${\displaystyle x}$ gets close to ${\displaystyle \infty }$. See Bemmaor and Zheng ^[5] for further analysis.

Cumulative distribution function

The cumulative distribution function of the shifted Gompertz distribution is:

${\displaystyle F(x;b,\eta )=\left(1-e^{-bx}\right)e^{-\eta e^{-bx}}{\text{ for }}x\geq 0.\,}$

Equivalently,

${\displaystyle F(x;p,q)=\left(1-e^{-(p+q)x}\right)e^{-\ln(1+q/p)e^{-(p+q)x}}{\text{ for }}x\geq 0.\,}$

${\displaystyle =\left(1-e^{-(p+q)x}\right){(1+q/p)^{-e^{-(p+q)x}}}{\text{ for }}x\geq 0.\,}$

Shapes

The shifted Gompertz density function can take on different shapes depending on the values of the shape parameter ${\displaystyle \eta }$:

• ${\displaystyle 0<\eta \leq 0.5\,}$ the probability density function has its mode at 0.
• ${\displaystyle \eta >0.5\,}$ the probability density function has its mode at

${\displaystyle {\text{mode}}=-{\frac {\ln(z^{\star })}{b}}\,\qquad 0<z^{\star }<1}$

where ${\displaystyle z^{\star }\,}$ is the smallest root of

${\displaystyle \eta ^{2}z^{2}-\eta (3+\eta )z+\eta +1=0\,,}$

which is

${\displaystyle z^{\star }=[3+\eta -(\eta ^{2}+2\eta +5)^{1/2}]/(2\eta ).}$