The Modified Weibull family

#' The Modified Weibull distribution

Usage

MW(mu.link = "log", sigma.link = "log", nu.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.

Value

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

Details

The Modified Weibull distribution with parameters mu, sigma and nu has density given by

\(f(x) = \mu (\sigma + \nu x) x^(\sigma - 1) \exp(\nu x) \exp(-\mu x^(\sigma) \exp(\nu x)),\)

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

References

Almalki, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Lai, C. D., Xie, M., & Murthy, D. N. P. (2003). A modified Weibull distribution. IEEE Transactions on reliability, 52(1), 33-37.

Author

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

Examples

# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rMW(n=100, mu = 2, sigma = 1.5, nu = 0.2)

# Fitting the model
require(gamlss)

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family= 'MW',
               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) 
#>    1.935706 
exp(coef(mod, what='sigma'))
#> (Intercept) 
#>    1.512932 
exp(coef(mod, what='nu'))
#> (Intercept) 
#>   0.2942034 

# Example 2
# Generating random values under some model
n     <- 200
x1    <- rpois(n, lambda=2)
x2    <- runif(n)
mu    <- exp(3 -1 * x1)
sigma <- exp(2 - 2 * x2)
nu    <- 0.2
x     <- rMW(n=n, mu, sigma, nu)

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

coef(mod, what="mu")
#> (Intercept)          x1 
#>   3.0337995  -0.9488072 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>    2.096852   -2.077852 
coef(mod, what='nu')
#> (Intercept) 
#>   -2.612791