Shifted Gompertz distribution
Encyclopedia
The shifted Gompertz distribution is the distribution of the largest of two independent random variable
s one of which has an exponential distribution
with parameter b and the other has a Gumbel distribution with parameters
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).
of the shifted Gompertz distribution is:
where
is the scale parameter
and
is the shape parameter
of the shifted Gompertz distribution.
of the shifted Gompertz distribution is:
. It is more flexible than the Gumbel distribution.
:
varies according to a gamma distribution with shape parameter 
and scale parameter 
(mean = 
), the cumulative distribution function is Gamma/Shifted Gompertz (G/SG). When 
is equal to one, the G/SG reduces to the Bass model.
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s one of which has an exponential distribution
Exponential distribution
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...
with parameter b and the other has a Gumbel distribution with parameters
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 truncatedTruncated distribution
In statistics, a truncated distribution is a conditional distribution that results from restricting the domain of some other probability distribution. Truncated distributions arise in practical statistics in cases where the ability to record, or even to know about, occurrences is limited to values...
at zero, the labelling can be considered as accurate. It has been used as a model of adoption of innovations
Diffusion of innovations
Diffusion of Innovations is a theory that seeks to explain how, why, and at what rate new ideas and technology spread through cultures. Everett Rogers, a professor of rural sociology, popularized the theory in his 1962 book Diffusion of Innovations...
. It was proposed by Bemmaor (1994).
Probability density function
The probability density functionProbability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
of the shifted Gompertz distribution is:
where
is the scale parameterScale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...
and
is the shape parameterShape parameter
In probability theory and statistics, a shape parameter is a kind of numerical parameter of a parametric family of probability distributions.- Definition :...
of the shifted Gompertz distribution.
Cumulative distribution function
The cumulative distribution functionCumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
of the shifted Gompertz distribution is:
Properties
The shifted Gompertz distribution is right-skewed for all values of. 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:
-
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
If varies according to a gamma distribution with shape parameter and scale parameter (mean = ), the cumulative distribution function is Gamma/Shifted Gompertz (G/SG). When is equal to one, the G/SG reduces to the Bass model.See also
- Gumbel distribution
- Generalized extreme value distribution
- Mixture modelMixture modelIn statistics, a mixture model is a probabilistic model for representing the presence of sub-populations within an overall population, without requiring that an observed data-set should identify the sub-population to which an individual observation belongs...
- Bass model
- Gompertz distribution