The Marshall-Olkin Extended Weibull family

Usage

MOEW(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 MOEW distribution in the gamlss() function.

Details

The Marshall-Olkin Extended Weibull distribution with parameters mu, sigma and nu has density given by

\(f(x) = \frac{\mu \sigma \nu (\nu x)^{\sigma - 1} exp\{{-(\nu x )^{\sigma}}\}}{\{1-(1-\mu) exp\{{-(\nu x )^{\sigma}}\} \}^{2}},\)

for x > 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 .

M.E G, E.K A, R.A J (2005). “Marshall–Olkin extended weibull distribution and its application to censored data.” Journal of Applied Statistics, 32(10), 1025–1034. doi:10.1080/02664760500165008 .

Author

Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co

Examples

# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rMOEW(n=400, mu=0.5, sigma=0.7, nu=1)

# Fitting the model
require(gamlss)

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='MOEW',
              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) 
#>   0.2044536 
exp(coef(mod, what='sigma'))
#> (Intercept) 
#>   0.8763439 
exp(coef(mod, what='nu'))
#> (Intercept) 
#>   0.4559564 

# Example 2
# Generating random values under some model
n <- 500
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(-1.20 + 3 * x1)
sigma <- exp(0.84 - 2 * x2)
nu <- 1
x <- rMOEW(n=n, mu, sigma, nu)

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

coef(mod, what="mu")
#> (Intercept)          x1 
#>   -2.635139    4.631801 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>   0.6658179  -1.4127674 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>    0.707802