The Gamma Weibull family

Usage

GammaW(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 GammaW distribution in the gamlss() function.

Details

The Gamma Weibull distribution with parameters mu, sigma and nu has density given by

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

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

References

Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32–55. doi:10.1016/j.ress.2013.11.010 .

Stacy EW, others (1962). “A generalization of the gamma distribution.” The Annals of mathematical statistics, 33(3), 1187–1192.

Author

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

Examples

# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rGammaW(n=100, mu = 0.5, sigma = 2, nu=1)

# Fitting the model
require(gamlss)

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='GammaW',
              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) 
#> 0.0004761073 
exp(coef(mod, what='sigma'))
#> (Intercept) 
#>    8.534264 
exp(coef(mod, what='nu'))
#> (Intercept) 
#>   0.2506652 

# Example 2
# Generating random values under some model
n     <- 200
x1    <- runif(n)
x2    <- runif(n)
mu    <- exp(-1.6 * x1)
sigma <- exp(1.1 - 1 * x2)
nu    <- 1
x     <- rGammaW(n=n, mu, sigma, nu)

mod <- gamlss(x~x1, mu.fo=~x1, sigma.fo=~x2, nu.fo=~1, family=GammaW,
              control=gamlss.control(n.cyc=50000, trace=FALSE))

coef(mod, what="mu")
#> (Intercept)          x1 
#>   -1.899445   -2.359121 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>   1.7900635  -0.9977632 
coef(mod, what='nu')
#> (Intercept) 
#>  -0.4255001