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

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

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

The KumIW distribution with \(\nu=1\) corresponds with the IW distribution.

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.” Pakistan journal of statistics and operation research, 8(3), 479–489.

Author

Freddy Hernandez, fhernanb@unal.edu.co

Examples

# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rKumIW(n=100, mu=1.5, sigma=2.3, nu=1)

# 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) 
#>    1.475667 
exp(coef(mod, what="sigma"))
#> (Intercept) 
#>    2.126994 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>    1.056507 

# Example 2
# Generating random values under some model
n <- 200
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(1 +  -1 * x1)
sigma <- exp(1 + -1 * x2)
nu <- 5
y <- rKumIW(n=n, mu=mu, sigma=sigma, nu=nu)

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

coef(mod, what="mu")
#> (Intercept)          x1 
#>   0.7880734  -1.4205981 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>   1.2452212  -0.9949117 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>    2.604682