MOK {RelDists}

The Marshall-Olkin Kappa family

Description

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.

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.

Value

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

Author(s)

Johan David Marin Benjumea, [e-mail]

References

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

See Also

dMOK

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'))
exp(coef(mod, what='sigma'))
exp(coef(mod, what='nu'))
exp(coef(mod, what='tau'))

# 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")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
exp(coef(mod, what="tau"))