The Extended Odd Frechet-Nadarjad-Hanhighi family

Usage

EOFNH(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 EOFNH distribution in the gamlss() function.

Details

The Extended Odd Frechet-Nadarjad-Hanhighi distribution with parameters mu, sigma, nu and tau has density given by

\(f(x)= \frac{\mu\sigma\nu\tau(1+\nu x)^{\sigma-1}e^{(1-(1+\nu x)^\sigma)}[1-(1-e^{(1-(1+\nu x)^\sigma)})^{\mu}]^{\tau-1}}{(1-e^{(1-(1+\nu x)^{\sigma})})^{\mu\tau+1}} e^{-[(1-e^{(1-(1+\nu x)^\sigma)})^{-\mu}-1]^{\tau}},\)

for \(x > 0\), \(\mu > 0\), \(\sigma > 0\), \(\nu > 0\) and \(\tau > 0\).

References

Nasiru S (2018). “Extended Odd Fréchet-G Family of Distributions.” Journal of Probability and Statistics, 2018.

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 <- rEOFNH(n=100, mu=1, sigma=2.1, nu=0.8, tau=1)

# Fitting the model
require(gamlss)

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, tau.fo=~1, family=EOFNH,
              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) 
#>   0.7709808 
exp(coef(mod, what='sigma'))
#> (Intercept) 
#>    1.950042 
exp(coef(mod, what='nu'))
#> (Intercept) 
#>   0.6955285 
exp(coef(mod, what='tau'))
#> (Intercept) 
#>    1.282222 

# 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 <- rEOFNH(n=n, mu, sigma, nu, tau)

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

coef(mod, what="mu")
#> (Intercept)          x1 
#>   0.8764594   1.1389432 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>    1.028776    1.978214 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>   0.4979135 
exp(coef(mod, what="tau"))
#> (Intercept) 
#>    0.336711