RelDists (version 1.0.2)

BS11: The Birnbaum-Saunders family - Santos-Neto et al. (2012) (P9 Based on the second Tweedie)

Description

The function BS11() 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

BS11(mu.link = "log", sigma.link = "log")

Value

Returns a gamlss.family object which can be used to fit a BS11 distribution in the gamlss() function.

Arguments

mu.link

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

sigma.link

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

Author

David Villegas Ceballos,

Details

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

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

for \(x>0\), \(\mu>0\) and \(\sigma>0\). In this parameterization, \(E(X) = \mu + \frac{\mu^2}{2\sigma}\) and \(Var(X) = \frac{\mu^3}{\sigma} + \frac{5\mu^4}{4\sigma^2}\).

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

dBS11.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(1234)
y <- rBS11(n=100, mu=1, sigma=12)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS11)

# 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 ~ BS11
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(0.5 - 1 * x1)      # Aprox 1
  sigma <- exp(1.9 + 1.2 * x2)   # Aprox 12
  y <- rBS11(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=BS11, data=dat)

summary(mod2)