RelDists (version 1.0.2)

BS12: The Birnbaum-Saunders family - Santos-Neto et al. (2012) (P10 Based on the third Tweedie)

Description

The function BS12() defines the Birnbaum-Saunders distribution, a two-parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS12(mu.link = "log", sigma.link = "log")

Value

Returns a gamlss.family object which can be used to fit a BS12 distribution in the gamlss() function.

Arguments

mu.link

defines the mu.link, with "log" link as the default for the mu parameter (representing the scale \(\beta\)).

sigma.link

defines the sigma.link, with "log" link as the default for the sigma parameter (representing the shape \(\psi\)).

Author

David Villegas Ceballos,

Details

The Birnbaum-Saunders distribution with parameters mu and sigma (where mu represents \(\beta\) and sigma represents \(\psi\)) has density given by

\(f(x|\mu,\sigma) = \frac{1}{\sqrt{2\pi}} \exp\left( -\frac{\sigma}{2} \left[ \frac{x}{\mu} + \frac{\mu}{x} - 2 \right] \right) \frac{[x+\mu]\sqrt{\sigma}}{2\sqrt{\mu x^3}}\)

for \(x>0\), \(\mu>0\) and \(\sigma>0\). In this parameterization, \(E(X) = \mu + \frac{\mu}{2\sigma}\) and \(Var(X) = \frac{\mu^2}{\sigma} + \frac{5\mu^2}{4\sigma^2}\).

References

Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., & Ahmed, S. E. (2012). On new parameterizations of the Birnbaum-Saunders distribution. Pakistan Journal of Statistics, 28(1), 1-26.

See Also

dBS12.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(12345)
y <- rBS12(n=100, mu=1, sigma=30)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS12)

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))

# Example 2
# Generating random values for a regression model

# A function to simulate a data set with Y ~ BS12
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(0.5 - 1 * x1)      # Aprox 1
  sigma <- exp(2.2 + 2.4 * x2)   # Aprox 30
  y <- rBS12(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(123)
dat <- gendat(n=200)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS12, data=dat)

summary(mod2)