# ShiftedGompertzDistribution

represents a shifted Gompertz distribution with scale parameter λ and shape parameter ξ.

# Details • ShiftedGompertzDistribution has been used to model technology adoption over time.
• The cumulative distribution function for value x in a shifted Gompertz distribution is given by for , and is zero for .
• ShiftedGompertzDistribution allows λ and ξ to be any positive real numbers.
• ShiftedGompertzDistribution allows λ to be a quantity of any unit dimension, and ξ to be a dimensionless quantity. »
• ShiftedGompertzDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Examples

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## Basic Examples(4)

Probability density function of a shifted Gompertz distribution:

Cumulative distribution function of a shifted Gompertz distribution:

Mean and variance of a shifted Gompertz distribution:

Median of a shifted Gompertz distribution:

## Scope(8)

Generate a sample of pseudorandom numbers from a shifted Gompertz distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Skewness of a shifted Gompertz distribution depends only on the shape parameter:

Kurtosis of a shifted Gompertz distribution depends only on the shape parameter:

Moment of a shifted Gompertz distribution:

Moment generating function and characteristic function:

Central moment:

Central moment generating function:

Factorial moment:

Factorial moment generating function:

Cumulant:

Cumulant generating function:

Hazard function of a shifted Gompertz distribution:

Quantile function of a shifted Gompertz distribution:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the median time:

## Applications(3)

The shifted Gompertz distribution can be used to model the growth and decline of interest in social networksfor example, Google search relative weekly counts for Facebook:

Fitting the data to a truncated shifted Gompertz distribution:

Compare the predictions from the model to the data:

Shifted Gompertz distribution is used to model time to technology adoption with rate of adoption λ and parameter ξ, related to propensity to adopt:

Median time to adoption in this model increases with ξ and decreases with rate λ:

Hazard rate of adoption is an increasing function of technology penetration level in this model:

For a heterogeneous population with respect to propensity to adopt, time to adoption is described by a ParameterMixtureDistribution. Exponential mixing distribution reproduces the classical Bass model:

In Bass model, the hazard function is linear in the technology penetration level (CDF):

Mixing parameter ξ with GammaDistribution gives gamma-shifted Gompertz model:

Conditional probability on the time for technology adoption for gamma-shifted Gompertz model:

Compare with the shifted Gompertz model:

The gamma-shifted Gompertz model (GSG) is used as a model for adoption of innovationsfor example, the number of adoptions of mammography scanners in consecutive time intervals:

Estimate the parameters by considering a multinomial distribution for the binned data and maximizing its LogLikelihood:

Estimate parameters using extended WorkingPrecision to avoid numerical instabilities, and raising the maximum number of iterations to ensure convergence:

Compare the predictions from the model to the data:

## Properties & Relations(3)

The maximum of ExponentialDistribution and ExtremeValueDistribution follows ShiftedGompertzDistribution:

In the limit of small shape parameter ξ, the shifted Gompertz distribution converges to exponential distribution with rate λ:

Moments of shifted Gompertz distribution for large shape parameter ξ are well approximated by moments of extreme value distribution:

Compare means:

Compare higher-order cumulants: