The Marshall-Olkin Kappa family

Usage

MOK(mu.link = "log", sigma.link = "log", nu.link = "log", tau.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.
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.

References

Javed M, Nawaz T, Irfan M (2018). “The Marshall-Olkin kappa distribution: properties and applications.” Journal of King Saud University-Science.

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) 
#>   0.9519014 
exp(coef(mod, what='sigma'))
#> (Intercept) 
#>    4.200663 
exp(coef(mod, what='nu'))
#> (Intercept) 
#>    3.152456 
exp(coef(mod, what='tau'))
#> (Intercept) 
#>   0.9140615 

# 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 
#>   -1.007821    4.815104 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>   0.7572979   1.6223883 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>   0.9772479 
exp(coef(mod, what="tau"))
#> (Intercept) 
#>    0.384337