Shifted Gompertz distribution
From Wikipedia, the free encyclopedia
Probability density function |
|
Cumulative distribution function |
|
Parameters | b > 0 scale (real) η > 0 shape (real) |
---|---|
Support | |
Probability density function (pdf) | |
Cumulative distribution function (cdf) | |
Mean |
where and |
Median | |
Mode | for , for where |
Variance |
where and |
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.
Contents |
[edit] Specification
[edit] Probability density function
The probability density function of the shifted Gompertz distribution is:
where b > 0 is the scale parameter and η > 0 is the shape parameter of the shifted Gompertz distribution.
[edit] Cumulative distribution function
The cumulative distribution function of the shifted Gompertz distribution is:
[edit] Properties
The shifted Gompertz distribution is right-skewed for all values of η.
[edit] 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
[edit] 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.
[edit] See also
[edit] 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.