The shifted Gompertz distribution is the distribution of the larger of two independent
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s one of which has an
exponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
with parameter
and the other has a
Gumbel distribution
In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.
Thi ...
with parameters
and
. 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, the labelling can be considered as accurate. It has been used as a model of
adoption of innovations. It was proposed by Bemmaor (1994). Some of its statistical properties have been studied further by Jiménez and Jodrá (2009)
and Jiménez Torres (2014).
It has been used to predict the growth and decline of social networks and on-line services and shown to be superior to the Bass model and Weibull distribution (Bauckhage and
Kersting Kersting is a surname. Notable people with the name include:
* Anthony F. Kersting (1916-2008), British architectural photographer
* Georg Friedrich Kersting (1785–1847), German painter
* Sachiko Furuhata-Kersting (born 1975), Japanese concert pi ...
2014).
Specification
Probability density function
The
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
of the shifted Gompertz distribution is:
:
where
is a
scale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution.
Definition
If a family o ...
and
is a
shape parameter
In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP.
...
. In the context of diffusion of innovations,
can be interpreted as the overall appeal of the innovation and
is the propensity to adopt in the propensity-to-adopt paradigm. The larger
is, the stronger the appeal and the larger
is, the smaller the propensity to adopt.
The distribution can be reparametrized according to the external versus internal influence paradigm with
as the coefficient of external influence and
as the coefficient of internal influence. Hence:
:
:
When
, the shifted Gompertz distribution reduces to an exponential distribution. When
, the proportion of adopters is nil: the innovation is a complete failure. The shape parameter of the probability density function is equal to
. Similar to the Bass model, the hazard rate
is equal to
when
is equal to
; it approaches
as
gets close to
. See Bemmaor and Zheng for further analysis.
Cumulative distribution function
The
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
of the shifted Gompertz distribution is:
:
Equivalently,
:
:
Properties
The shifted Gompertz distribution is right-skewed for all values of
. It is more flexible than the
Gumbel distribution
In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.
Thi ...
. The hazard rate is a concave function of
which increases from
to
: its curvature is all the steeper as
is large. In the context of the diffusion of innovations, the effect of word of mouth (i.e., the previous adopters) on the likelihood to adopt decreases as the proportion of adopters increases. (For comparison, in the Bass model, the effect remains the same over time). The parameter
captures the growth of the hazard rate when
varies from
to
.
Shapes
The shifted Gompertz density function can take on different shapes depending on the values of the shape parameter
:
*
the probability density function has its mode at 0.
*
the probability density function has its mode at
::
:where
is the smallest root of
::
:which is
::
Related distributions
When
varies according to a
gamma distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...
with shape parameter
and scale parameter
(mean =
), the distribution of
is Gamma/Shifted Gompertz (G/SG). When
is equal to one, the G/SG reduces to the
Bass model (Bemmaor 1994). The three-parameter G/SG has been applied by Dover, Goldenberg and Shapira (2009) and Van den Bulte and Stremersch (2004) among others in the context of the diffusion of innovations. The model is discussed in Chandrasekaran and Tellis
[
](2007).Similar to the shifted Gompertz distribution, the G/SG can either be represented according to the propensity-to-adopt paradigm or according to the innovation-imitation paradigm. In the latter case, it includes three parameters:
and
with
and
. The parameter
modifies the curvature of the hazard rate as expressed as a function of
: when
is less than 0.5, it decreases to a minimum prior to increasing at an increasing rate as
increases, it is convex when
is less than one and larger or equal to 0.5, linear when
is equal to one, and concave when
is larger than one. Here are some special cases of the G/SG distribution in the case of homogeneity (across the population) with respect to the likelihood to adopt at a given time:
= Exponential
= Left-skewed two-parameter distribution
= Bass model
= Shifted Gompertz
with:
One can compare the parameters
and
across the values of
as they capture the same notions. In all the cases, the hazard rate is either constant or a monotonically increasing function of
(positive word of mouth). As the diffusion curve is all the more skewed as
becomes large, we expect
to decrease as the level of right-skew increases.
See also
*
Gumbel distribution
In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.
Thi ...
*
Generalized extreme value distribution
In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known ...
*
Mixture model
In statistics, a mixture model is a probabilistic model for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation ...
*
Bass model
*
Gompertz distribution
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 actu ...
References
{{ProbDistributions, continuous-semi-infinite
Continuous distributions