The Kumaraswamy Inverse Weibull family

Usage

KumIW(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 KumIW distribution in the gamlss() function.

Details

The Kumaraswamy Inverse Weibull Distribution with parameters mu, sigma and nu has density given by

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

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

References

Almalki, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Shahbaz, M. Q., Shahbaz, S., & Butt, N. S. (2012). The Kumaraswamy Inverse Weibull Distribution. Pakistan journal of statistics and operation research, 479-489.

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 <- rKumIW(n=100, mu = 1.5, sigma=  1.5, nu = 5)

# Fitting the model
require(gamlss)

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family="KumIW",
              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) 
#>    1.237975 
exp(coef(mod, what="sigma"))
#> (Intercept) 
#>     1.24013 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>    3.757141 

# 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(1 - x1)
sigma <- exp(1 - x2)
nu <- 5
x <- rKumIW(n=n, mu, sigma, nu)

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

coef(mod, what="mu")
#> (Intercept)          x1 
#>  -0.2206343   0.9095627 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>   0.8028841  -0.9045411 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>     3.84521