The Marshall-Olkin Kappa 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.
- tau.link
defines the tau.link, with "log" link as the default for the tau parameter.
Value
Returns a gamlss.family object which can be used to fit a MOK distribution in the gamlss() function.
Details
The Marshall-Olkin Kappa distribution with parameters mu,
sigma, nu and tau has density given by
\(f(x)=\frac{\tau\frac{\mu\nu}{\sigma}\left(\frac{x}{\sigma}\right)^{\nu-1} \left(\mu+\left(\frac{x}{\sigma}\right)^{\mu\nu}\right)^{-\frac{\mu+1}{\mu}}}{\left(\tau+(1-\tau)\left(\frac{\left(\frac{x}{\sigma}\right)^{\mu\nu}}{\mu+\left(\frac{x}{\sigma}\right)^{\mu\nu}}\right)^{\frac{1}{\mu}}\right)^2}\)
for x > 0.
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 <- rMOK(n=100, mu = 1, sigma = 3.5, nu = 3, tau = 2)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, tau.fo=~1, family=MOK,
control=gamlss.control(n.cyc=5000, trace=FALSE))
# Extracting the fitted values for mu, sigma, nu and tau
# using the inverse link function
exp(coef(mod, what='mu'))
#> (Intercept)
#> 1.80385
exp(coef(mod, what='sigma'))
#> (Intercept)
#> 5.990718
exp(coef(mod, what='nu'))
#> (Intercept)
#> 2.301633
exp(coef(mod, what='tau'))
#> (Intercept)
#> 0.4341955
# 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(0.5 + x1)
sigma <- exp(0.8 + x2)
nu <- 1
tau <- 0.5
x <- rMOK(n=n, mu, sigma, nu, tau)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, tau.fo=~1, family=MOK,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
#> (Intercept) x1
#> 1.865873 -1.716180
coef(mod, what="sigma")
#> (Intercept) x2
#> 0.8279058 -0.5062949
exp(coef(mod, what="nu"))
#> (Intercept)
#> 0.7884017
exp(coef(mod, what="tau"))
#> (Intercept)
#> 1.466607