Shifted Gompertz distribution
|Probability density function
|Cumulative distribution function
|Parameters|| scale (real)
The shifted Gompertz distribution is the distribution of the largest of two independent random variables 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).
Probability density function
The probability density function of the shifted Gompertz distribution is:
Cumulative distribution function
The cumulative distribution function of the shifted Gompertz distribution is:
The shifted Gompertz distribution is right-skewed for all values of . It is more flexible than the Gumbel distribution.
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
If varies according to a gamma distribution 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.
- Gumbel distribution
- Generalized extreme value distribution
- Mixture model
- Bass model
- Gompertz distribution
||This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (April 2012)|
- 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. pp. 201–223. ISBN 0-7923-9388-0.
- Chandrasekaran, Deepa; Tellis, Gerard J. (2007). "A Critical Review of Marketing Research on Diffusion of New Products". In Naresh K. Malhotra. Review of Marketing Research 3. Armonk: M.E. Sharpe. pp. 39–80. ISBN 978-0-7656-1306-6.
- Dover, Yaniv; Goldenberg, Jacob; Shapira, Daniel (2012). "Network Traces on Penetration: Uncovering Degree Distribution From Adoption Data". Marketing Science. doi:10.1287/mksc.1120.0711.
- Jimenez, Fernando; Jodra, Pedro (2009). "A Note on the Moments and Computer Generation of the Shifted Gompertz Distribution". Communications in Statistics - Theory and Methods 38 (1): 78–89. doi:10.1080/03610920802155502.
- Van den Bulte, Christophe; Stremersch, Stefan (2004). "Social Contagion and Income Heterogeneity in New Product Diffusion: A Meta-Analytic Test". Marketing Science 23 (4): 530–544. doi:10.1287/mksc.1040.0054.