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Shifted Gompertz
Probability density function
Probability density plots of shifted Gompertz distributions
Cumulative distribution function
Cumulative distribution plots of shifted Gompertz distributions
Parameters scale (real)
shape (real)
Support
PDF
CDF
Mean

where and

Mode for , for where
Variance

where and


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 respectively. It has been used as a model of adoption of innovation.

Specification

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Probability density function

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The probability density function of the shifted Gompertz distribution is:

where is the scale parameter and is the shape parameter of the shifted Gompertz distribution.

Cumulative distribution function

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The cumulative distribution function of the shifted Gompertz distribution is:


Properties

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

Shapes

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

  • the probability density function has mode 0.
  • the probability density function has the mode at where is the smallest root of which is
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If varies according to a gamma distribution with shape parameter and scale parameter (mean = ), the cumulative distribution function is Gamma/Shifted Gompertz.

See also

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References

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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 (ed.), Research Traditions in Marketing, Boston: Kluwer Academic Publishers, pp. 201–223. Van Den Bulte, Christophe (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. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) Chandrasekaran, Deepa; Tellis, Gerard J. (2007), "A Critical Review of Marketing Research on Diffusion of New Products", in Naresh K. Malhotra (ed.), Review of Marketing Research, vol. 3, Armonk: M.E. Sharpe.