The Extended Weibull family

Usage

ExW(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 ExW distribution in the gamlss() function.

Details

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

\(f(x) = \frac{\mu \sigma \nu x^{\sigma -1} exp({-\mu x^{\sigma}})} {[1 -(1-\nu) exp({-\mu x^{\sigma}})]^2},\)

for x > 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 .

Tieling Z, Min X (2007). “Failure Data Analysis with Extended Weibull Distribution.” Communications in Statistics - Simulation and Computation, 36, 579–592. doi:10.1080/03610910701236081 .

Author

Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co

Examples

# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rExW(n=200, mu=0.3, sigma=2, nu=0.05)

# Fitting the model
require(gamlss)

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='ExW',
              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) 
#>    2.957026 
exp(coef(mod, what='sigma'))
#> (Intercept) 
#>    1.225708 
exp(coef(mod, what='nu'))
#> (Intercept) 
#>     1.94751 

# Example 2
# Generating random values under some model
n <- 500
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(-2 + 3 * x1)
sigma <- exp(1.3 - 2 * x2)
nu <- 0.05
x <- rExW(n=n, mu, sigma, nu)

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

coef(mod, what="mu")
#> (Intercept)          x1 
#>   1.1367679   0.4107476 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>    1.084149   -2.774546 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>    1.938714