The Generalized Gompertz family

Usage

GGD(mu.link = "log", sigma.link = "log", nu.link = "log")

Arguments

mu.link: defines the mu.link, with "log" link as the default for the mu parameter.
sigma.link: defines the sigma.link, with "log" link as the default for the sigma.
nu.link: defines the nu.link, with "log" link as the default for the nu parameter.

Value

Returns a gamlss.family object which can be used to fit a GGD distribution in the gamlss() function. .

Details

The Generalized Gompertz Distribution with parameters mu, sigma and nu has density given by

\(f(x)= \nu \mu \exp(-\frac{\mu}{\sigma}(\exp(\sigma x - 1))) (1 - \exp(-\frac{\mu}{\sigma}(\exp(\sigma x - 1))))^{(\nu - 1)} ,\)

for \(x \geq 0\), \(\mu > 0\), \(\sigma \geq 0\) and \(\nu > 0\)

References

El-Gohary A, Alshamrani A, Al-Otaibi AN (2013). “The generalized Gompertz distribution.” Applied Mathematical Modelling, 37(1-2), 13–24.

Author

Johan David Marin Benjumea, johand.marin@udea.edu.co

Examples

#Example 1
# Generating some random values with
# known mu, sigma, nu and tau
y <- rGGD(n=1000, mu=1, sigma=0.3, nu=1.5)

# Fitting the model
require(gamlss)

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='GGD',
              control=gamlss.control(n.cyc=5000, trace=FALSE))

# Extracting the fitted values for mu, sigma and nu 
# using the inverse link function
exp(coef(mod, what='mu'))
#> (Intercept) 
#>   0.9945394 
exp(coef(mod, what='sigma'))
#> (Intercept) 
#>   0.3118921 
exp(coef(mod, what='nu'))
#> (Intercept) 
#>     1.51825 

# Example 2
# Generating random values under some model
n <- 200
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(0.5 - x1)
sigma <- exp(-1 - x2)
nu <- 1.5
x <- rGGD(n=n, mu, sigma, nu)

mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, family=GGD,
              control=gamlss.control(n.cyc=5000, trace=FALSE))

coef(mod, what="mu")
#> (Intercept)          x1 
#>   0.1230266  -0.1127540 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>   -2.686041    1.831254 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>    1.574023