The Exponentiated Modifien Weibull Extension family

Usage

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

Value

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

Details

The Beta-Weibull distribution with parameters mu, sigma, nu and tau has density given by

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

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

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 .

Sarhan AM, Apaloo J (2013). “Exponentiated modified Weibull extension distribution.” Reliability Engineering & System Safety, 112, 137--144.

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 <- rEMWEx(n=100, mu = 1, sigma =1.21, nu=1, tau=2)

# Fitting the model
require(gamlss)

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

# Extracting the fitted values for mu, sigma, nu and tau
# using the inverse link function
exp(coef(mod, what='mu'))
#> (Intercept) 
#>   0.9793743 
exp(coef(mod, what='sigma'))
#> (Intercept) 
#>    1.092549 
exp(coef(mod, what='nu'))
#> (Intercept) 
#>    1.321336 
exp(coef(mod, what='tau'))
#> (Intercept) 
#>    2.792185 

# 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(0.75 - x1)
sigma <- exp(0.5 - x2)
nu <- 1
tau <- 2
x <- rEMWEx(n=n, mu, sigma, nu, tau)

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

coef(mod, what="mu")
#> (Intercept)          x1 
#>  0.56343941  0.05134896 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>   0.6517826   0.2681013 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>    1.000626 
exp(coef(mod, what="tau"))
#> (Intercept) 
#>   0.7482058