The Marshall-Olkin Extended Weibull family
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)
#> 0.5766113
exp(coef(mod, what='sigma'))
#> (Intercept)
#> 0.6707689
exp(coef(mod, what='nu'))
#> (Intercept)
#> 1.072079
# 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
#> -1.287091 4.307125
coef(mod, what="sigma")
#> (Intercept) x2
#> 0.966983 -2.462201
exp(coef(mod, what="nu"))
#> (Intercept)
#> 1.501479