The Beta Generalized Exponentiated family

Usage

BGE(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 BGE distribution in the gamlss() function.

Details

The Beta Generalized Exponentiated distribution with parameters mu, sigma, nu and tau has density given by

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

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

References

Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32--55. doi:10.1016/j.ress.2013.11.010 .

Barreto-Souza W, Santos AH, Cordeiro GM (2010). “The beta generalized exponential distribution.” Journal of Statistical Computation and Simulation, 80(2), 159--172.

Author

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

Examples

# Generating some random values with
# known mu, sigma, nu and tau
y <- rBGE(n=100, mu = 1.5, sigma =1.7, nu=1, tau=1)

# Fitting the model
require(gamlss)
#> Loading required package: gamlss
#> Loading required package: splines
#> Loading required package: gamlss.data
#> 
#> Attaching package: ‘gamlss.data’
#> The following object is masked from ‘package:datasets’:
#> 
#>     sleep
#> Loading required package: gamlss.dist
#> Loading required package: MASS
#> Loading required package: nlme
#> 
#> Attaching package: ‘nlme’
#> The following object is masked from ‘package:BBmisc’:
#> 
#>     collapse
#>  **********   GAMLSS Version 5.4-10  ********** 
#> For more on GAMLSS look at https://www.gamlss.com/
#> Type gamlssNews() to see new features/changes/bug fixes.

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, tau.fo=~1, family=BGE,
              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) 
#>    1.042086 
exp(coef(mod, what='sigma'))
#> (Intercept) 
#>    1.956497 
exp(coef(mod, what='nu'))
#> (Intercept) 
#>   0.9755567 
exp(coef(mod, what='tau'))
#> (Intercept) 
#>    1.452975 

# 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 <- 1
x <- rBGE(n=n, mu, sigma, nu, tau)

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

coef(mod, what="mu")
#> (Intercept)          x1 
#>   1.4478803  -0.9431118 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>   1.7208372  -0.7308966 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>   0.3488586 
exp(coef(mod, what="tau"))
#> (Intercept) 
#>   0.4994917