The Weibull Geometric family

The Weibull Geometric distribution

Usage

WG(mu.link = "log", sigma.link = "log", nu.link = "logit")

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 WG distribution in the gamlss() function.

Details

The weibull geometric distribution with parameters mu, sigma and nu has density given by

\(f(x) = (\sigma \mu^\sigma (1-\nu) x^(\sigma - 1) \exp(-(\mu x)^\sigma)) (1- \nu \exp(-(\mu x)^\sigma))^{-2},\)

for \(x > 0\), \(\mu > 0\), \(\sigma > 0\) and \(0 < \nu < 1\).

References

Barreto-Souza W, de Morais AL, Cordeiro GM (2011). “The Weibull-geometric distribution.” Journal of Statistical Computation and Simulation, 81(5), 645–657.

Author

Johan David Marin Benjumea, johand.marin@udea.edu.co

Examples

# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rWG(n=100,  mu = 0.9, sigma = 2, nu = 0.5)

# Fitting the model
require(gamlss)

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='WG',
              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) 
#>    2.047776 
exp(coef(mod, what='sigma'))
#> (Intercept) 
#>   0.9891653 
exp(coef(mod, what='nu'))
#> (Intercept) 
#>    1.439952 

# Example 2
# Generating random values under some model
n     <- 200
x1    <- runif(n)
x2    <- runif(n)
mu    <- exp(- 0.2 * x1)
sigma <- exp(1.2 - 1 * x2)
nu    <- 0.5
x     <- rWG(n=n, mu, sigma, nu)

mod <- gamlss(x~x1, mu.fo=~x1, sigma.fo=~x2, nu.fo=~1, family=WG,
              control=gamlss.control(n.cyc=50000, trace=FALSE))

coef(mod, what="mu")
#> (Intercept)          x1 
#>   0.9340655  -0.2575587 
coef(mod, what="sigma")
#> (Intercept)          x2 
#> -0.18783559  0.04320311 
coef(mod, what='nu')
#> (Intercept) 
#>  -0.6876997