In probability and statistics, the Gompertz distribution is a continuous probability distribution, named after Benjamin Gompertz. The Gompertz distribution is often applied to describe the distribution of adult lifespans by demographers and actuaries. Related fields of science such as biology and gerontology also considered the Gompertz distribution for the analysis of survival. More recently, computer scientists have also started to model the failure rates of computer code by the Gompertz distribution. In marketing, it has been used as an individual-level simulation for customer lifetime value modeling. In network theory, particularly the Erdős–Rényi model, the walk length of a random self-avoiding walk (SAW) is distributed according to the Gompertz distribution.
Contents
Specification
Probability density function
The probability density function of the Gompertz distribution is:
f
(
x
;
η
,
b
)
=
b
η
exp
(
η
+
b
x
−
η
e
b
x
)
for
x
≥
0
,
{\displaystyle f\left(x;\eta ,b\right)=b\eta \exp \left(\eta +bx-\eta e^{bx}\right){\text{for }}x\geq 0,\,}
where
b
>
0
Cumulative distribution function
The cumulative distribution function of the Gompertz distribution is:
F
(
x
;
η
,
b
)
=
1
−
exp
(
−
η
(
e
b
x
−
1
)
)
,
{\displaystyle F\left(x;\eta ,b\right)=1-\exp \left(-\eta \left(e^{bx}-1\right)\right),}
where
η
,
b
>
0
,
{\displaystyle \eta ,b>0,}
Moment generating function
The moment generating function is:
E
(
e
−
t
X
)
=
η
e
η
E
t
/
b
(
η
)
{\displaystyle {\text{E}}\left(e^{-tX}\right)=\eta e^{\eta }{\text{E}}_{t/b}\left(\eta \right)}
where
E
t
/
b
(
η
)
=
∫
1
∞
e
−
η
v
Properties
The Gompertz distribution is a flexible distribution that can be skewed to the right and to the left. Its hazard function
h
(
x
)
=
η
b
e
b
x
{\displaystyle h(x)=\eta be^{bx}}
is a convex function of
F
(
x
;
η
,
b
)
{\displaystyle F\left(x;\eta ,b\right)}
. The model can be fitted into the innovation-imitation paradigm with
p
=
η
b
{\displaystyle p=\eta b}
as the coefficient of innovation and
b
{\displaystyle b}
Shapes
The Gompertz density function can take on different shapes depending on the values of the shape parameter
η
{\displaystyle \eta \,\!}
:
When
η
≥
1
,
{\displaystyle \eta \geq 1,\,}
the probability density function has its mode at 0.
When
0
<
η
<
1
,
{\displaystyle 0<\eta <1,\,}
the probability density function has its mode at
x
∗
=
(
1
/
b
)
ln
(
1
/
Kullback–Leibler divergence
If
f
1
{\displaystyle f_{1}}
and
f
2
{\displaystyle f_{2}}
are the probability density functions of two Gompertz distributions, then their Kullback–Leibler divergence is given by
D
K
L
(
f
1
∥
f
2
)
=
∫
0
∞
f
1
(
x
;
b
1
,
η
1
)
ln
Related distributions
If X is defined to be the result of sampling from a Gumbel distribution until a negative value Y is produced, and setting X=−Y, then X has a Gompertz distribution.
The gamma distribution is a natural conjugate prior to a Gompertz likelihood with known scale parameter
b
.
{\displaystyle b\,\!.}
When
η
{\displaystyle \eta \,\!}
varies according to a gamma distribution with shape parameter
α
{\displaystyle \alpha \,\!}
and scale parameter
β
{\displaystyle \beta \,\!}
(mean =
α
/
β
{\displaystyle \alpha /\beta \,\!}
), the distribution of
x
{\displaystyle x}
is Gamma/Gompertz.
If
Y
∼
Applications
In hydrology the Gompertz distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. The blue picture illustrates an example of fitting the Gompertz distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.