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 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 .

Shahbaz MQ, Shahbaz S, Butt NS (2012). “The Kumaraswamy-Inverse Weibull Distribution.” Shahbaz, MQ, Shahbaz, S., & Butt, NS (2012). The Kumaraswamy--Inverse Weibull Distribution. Pakistan journal of statistics and operation research, 8(3), 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=1000, 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) 
#>    1.028873 
exp(coef(mod, what='sigma'))
#> (Intercept) 
#>    1.511681 
exp(coef(mod, what='nu'))
#> (Intercept) 
#>    5.184874 

# 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 
#>  -0.3848230   0.5982428 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>    1.451138   -1.662854 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>    6.800247