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Encyclopedia > Shifted Gompertz Distribution
Shifted Gompertz
Probability density function
Probability density plots of shifted Gompertz distributions
Cumulative distribution function
Cumulative distribution plots of shifted Gompertz distributions
Parameters b > 0 scale (real)
η > 0 shape (real)
Support
Probability density function (pdf)
Cumulative distribution function (cdf)
Mean

where and In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In mathematics, the real numbers may be described informally in several different ways. ... In probability theory and statistics, a shape parameter is a special kind of numerical parameter of a parametric family of probability distributions. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

Median
Mode for , for where
Variance

where and In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function


The shifted Gompertz distribution is the distribution of the largest order statistic of two independent random variables which are distributed exponential and Gompertz with parameters b and b and η respectively. It has been used as a model of adoption of innovation. Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function In information theory, information entropy or Shannons entropy is a measure of the average information content associated with the outcome of a random variable. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... In statistics, the kth order statistic of a statistical sample is equal its kth-smallest value. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory and statistics the Gumbel distribution (named after Emil Julius Gumbel (1891–1966)) is used to find the minimum (or the maximum) of a number of samples of various distributions. ... The study of the diffusion of innovation is the study of how, why, and at what rate new ideas spread through cultures. ...

Contents

Specification

Probability density function

The probability density function of the shifted Gompertz distribution is: In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

where b > 0 is the scale parameter and η > 0 is the shape parameter of the shifted Gompertz distribution. In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In probability theory and statistics, a shape parameter is a special kind of numerical parameter of a parametric family of probability distributions. ...


Cumulative distribution function

The cumulative distribution function of the shifted Gompertz distribution is: In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

Properties

The shifted Gompertz distribution is right-skewed for all values of η.


Shapes

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

  • the probability density function has mode 0.
  • the probability density function has the mode at where is the smallest root of which is

Related Distributions

If η varies according to a gamma distribution with shape parameter α and scale parameter β (mean = αβ), the cumulative distribution function is Gamma/Shifted Gompertz. In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions that represents the sum of exponentially distributed random variables, each of which has mean . ...


See also

In probability theory and statistics the Gumbel distribution (named after Emil Julius Gumbel (1891–1966)) is used to find the minimum (or the maximum) of a number of samples of various distributions. ... In probability theory and statistics, the generalized extreme value distribution (GEV) is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. ... In mathematics, the term mixture model is a model in which independent variables are fractions of a total. ...

References

Bemmaor, Albert C. (1994), "Modeling the Diffusion of New Durable Goods: Word-of-Mouth Effect Versus Consumer Heterogeneity", in G. Laurent, G.L. Lilien & B. Pras, Research Traditions in Marketing, Boston: Kluwer Academic Publishers.


Van Den Bulte, Christophe; Stefan Stremersch (2004). "Social Contagion and Income Heterogeneity in New Product Diffusion: A Meta-Analytic Test". Marketing Science 23 (4): 530–544. 


Chandrasekaran, Deepa & Gerard J. Tellis (2007), "A Critical Review of Marketing Research on Diffusion of New Products", in Naresh K. Malhotra, Review of Marketing Research, vol. 3, Armonk: M.E. Sharpe.

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Miscellaneous: Cantorconditionalexponential familyinfinitely divisible • location-scale family • marginalmaximum entropy • phase-type • posteriorprior • quasi • samplingsingular

 
 

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