Shifted Gompertz distribution

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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

Variance

where and

The shifted Gompertz distribution is the distribution of the largest of two independent random variables one of which has an exponential distribution with parameter and the other has a Gumbel distribution with parameters and . 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, the labelling can be considered as accurate. It has been used as a model of adoption of innovations. It was proposed by Bemmaor[1] (1994). Some of its statistical properties have been studied further by Jiménez and Jodrá [2](2009) and Jiménez Torres [3](2014).

It has been used to predict the growth and decline of social networks and on-line services and shown to be superior to the Bass model and Weibull distribution (Bauckhage and Kersting [4] 2014).

Specification[edit]

Probability density function[edit]

The probability density function of the shifted Gompertz distribution is:

where is a scale parameter and is a shape parameter. In the context of diffusion of innovations, can be interpreted as the overall appeal of the innovation and is the propensity to adopt in the propensity-to-adopt paradigm. The larger is, the stronger the appeal and the larger is, the smaller the propensity to adopt.

The distribution can be reparametrized according to the innovation-imitation paradigm with as the coefficient of innovation and as the coefficient of imitation. Hence:

When , the shifted Gompertz distribution reduces to an exponential distribution. When , the proportion of adopters is nill: the innovation is a complete failure. The shape parameter becomes equal to .

Cumulative distribution function[edit]

The cumulative distribution function of the shifted Gompertz distribution is:

Equivalently,

Properties[edit]

The shifted Gompertz distribution is right-skewed for all values of . It is more flexible than the Gumbel distribution. The hazard function is a concave function of which increases from to : its curvature is all the steeper as is large. Contrary to some other distributions, the hazard function cannot be expressed as a closed-form expression of . In the context of the diffusion of innovations, the effect of word of mouth on the likelihood to adopt is stronger at the early stages of adoption than at the later ones. The parameter captures the growth of the hazard function from to .

Shapes[edit]

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[edit]

When 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 (Bemmaor 1994). The three-parameter G/SG has been applied by Dover, Goldenberg and Shapira [5](2009) and Van den Bulte and Stremersch [6](2004) among others in the context of the diffusion of innovations. The model is discussed in Chandrasekaran and Tellis [7](2007).Similar to the shifted Gompertz distribution, the G/SG can either be represented according to the propensity-to-adopt paradigm or according to the innovation-imitation paradigm. In the latter case, it includes three parameters: and with and . The parameter modifies the curvature of the hazard rate as expressed as a function of : it is convex when is less than one, linear when is equal to one, and concave when is larger than one. Similar to the shifted Gompertz distribution, there is no closed-form expression of the hazard function except when . Here are some special cases of the G/SG distribution in its innovation-imitation formulation:

                         = Exponential
                         = Competing two-parameter distribution
                          = Bass model
                         = Shifted Gompertz
 

with:

              

One can compare the parameters and across the values of as they capture the same notions. In all the cases, the hazard function is either constant or a monotonically increasing function of (positive word of mouth). As the diffusion curve is all the more skewed as becomes large, we expect to decrease as the level of skew increases.

See also[edit]

References[edit]

  1. ^ 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. 
  2. ^ Jiménez, Fernando; Jodrá, 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. 
  3. ^ Jiménez Torres, Fernando (2014). "Estimation of the Parameters of the Shifted Gompertz Distribution, Using Least Squares, Maximum Likelihood and Moments Methods". Journal of Computational and Applied Mathematics 255 (1): 867–877. doi:10.1016/j.cam.2013.07.004. 
  4. ^ Bauckhage, Christian; Kersting, Kristian (2014). "Strong Regularities in Growth and Decline of Popularity of Social Media Services". arXiv 1401.6529 [CS.SI]. 
  5. ^ Dover, Yaniv; Goldenberg, Jacob; Shapira, Daniel (2012). "Network Traces on Penetration: Uncovering Degree Distribution From Adoption Data". Marketing Science 31 (4): 689–712. doi:10.1287/mksc.1120.0711. 
  6. ^ 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. 
  7. ^ 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.