The Odd Weibull family

The function OW() defines the Odd Weibull distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

OW(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.

Value

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

Details

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

\(f(t) = \left( \frac{\sigma\nu}{t} \right) (\mu t)^\sigma e^{(\mu t)^\sigma} \left(e^{(\mu t)^{\sigma}}-1\right)^{\nu-1} \left[ 1 + \left(e^{(\mu t)^{\sigma}}-1\right)^\nu \right]^{-2}\)

for x > 0.

References

Cooray K (2006). “Generalization of the Weibull distribution: The odd Weibull family.” Statistical Modelling, 6(3), 265--277. ISSN 1471082X, doi:10.1191/1471082X06st116oa .

Author

Jaime Mosquera Gutiérrez jmosquerag@unal.edu.co

Examples

# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rOW(n=200, mu=0.1, sigma=7, nu = 1.1)

# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family="OW",
              control=gamlss.control(n.cyc=500, trace=FALSE))

# Extracting the fitted values for mu, sigma and nu
# using the inverse link function
exp(coef(mod, what='mu'))
#> (Intercept) 
#>   0.1005415 
exp(coef(mod, what='sigma'))
#> (Intercept) 
#>     7.76591 
exp(coef(mod, what='nu'))
#> (Intercept) 
#>   0.9671895 

# Example 2
# Generating random values under some model
n <- 500
x1 <- runif(n)
x2 <- runif(n)
x3 <- runif(n)
mu <- exp(1.2 + 2 * x1)
sigma <- 2.12 + 3 * x2
nu <- exp(0.2 - x3)
x <- rOW(n=n, mu, sigma, nu)

mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~x3, 
              family=OW(sigma.link='identity'), 
              control=gamlss.control(n.cyc=300, trace=FALSE))

coef(mod, what='mu')
#> (Intercept)          x1 
#>    1.208619    2.013742 
coef(mod, what='sigma')
#> (Intercept)          x2 
#>    2.016076    3.080777 
coef(mod, what='nu')
#> (Intercept)          x3 
#>   0.2525304  -1.1715029