# Shifted Gompertz distribution

Parameters Probability density function Cumulative distribution function $b>0$ scale (real) $\eta>0$ shape (real) $x \in [0, \infty)\!$ $b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]$ $\left(1 - e^{-bx}\right)e^{-\eta e^{-bx}}$ $(-1/b)\{\mathrm{E}[\ln(X)] - \ln(\eta)\}\,$ 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} $0 \text{ for }0 < \eta \leq 0.5$ $(-1/b)\ln(z^\star)\text{, for } \eta > 0.5$ $\text{ where }z^\star = [3 + \eta - (\eta^2 + 2\eta + 5)^{1/2}]/(2\eta)$ $(1/b^2)(\mathrm{E}\{[\ln(X)]^2\} - (\mathrm{E}[\ln(X)])^2)\,$ 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}

The shifted Gompertz distribution is the distribution of the largest of two independent random variables one of which has an exponential distribution with parameter b and the other has a Gumbel distribution with parameters $\eta$ and b. In its original formulation the distribution was expressed referring to the Gompertz distribution instead of the Gumbel distribution but, since the Gompertz distribution is a reverted Gumbel distribution (truncated at zero), the labelling can be considered as accurate. It has been used as a model of adoption of innovations. It was proposed by Bemmaor (1994).

## Specification

### Probability density function

The probability density function of the shifted Gompertz distribution is:

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

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:

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

## Properties

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

### Shapes

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

• $0 < \eta \leq 0.5\,$ the probability density function has its mode at 0.
• $\eta > 0.5\,$ the probability density function has its mode at
$\text{mode}=-\frac{\ln(z^\star)}{b}\, \qquad 0 < z^\star < 1$
where $z^\star\,$ is the smallest root of
$\eta^2z^2 - \eta(3 + \eta)z + \eta + 1 = 0\,,$
which is
$z^\star = [3 + \eta - (\eta^2 + 2\eta + 5)^{1/2}]/(2\eta).$

## Related distributions

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