The Generalized modified Weibull distribution
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 "sqrt" link as the default for the nu parameter.
- tau.link
defines the tau.link, with "sqrt" link as the default for the tau parameter.
Value
Returns a gamlss.family object which can be used to fit a GMW distribution in the gamlss() function.
Details
The Generalized modified Weibull distribution with parameters mu,
sigma, nu and tau has density given by
\(f(x)= \mu \sigma x^{\nu - 1}(\nu + \tau x) \exp(\tau x - \mu x^{\nu} e^{\tau x}) [1 - \exp(- \mu x^{\nu} e^{\tau x})]^{\sigma-1},\)
for x > 0.
Examples
# Example 1
# Generating some random values with
# known mu, sigma, nu and tau
y <- rGMW(n=100, mu=2, sigma=0.5, nu=2, tau=1.5)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, tau.fo=~ 1, family='GMW',
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)
#> 1.426891
exp(coef(mod, what='sigma'))
#> (Intercept)
#> 1.080822
(coef(mod, what='nu'))^2
#> (Intercept)
#> 1.0625
(coef(mod, what='tau'))^2
#> (Intercept)
#> 2.158165
# Example 2
# Generating random values under some model
if (FALSE) {
n <- 1000
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(2 + -3 * x1)
sigma <- exp(3 - 2 * x2)
nu <- 2
tau <- 1.5
x <- rGMW(n=n, mu, sigma, nu, tau)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, tau.fo=~ 1, family="GMW",
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
coef(mod, what="nu")^2
coef(mod, what="tau")^2
}