RelDists (version 1.0.2)

BS8: The Birnbaum-Saunders family - Santos-Neto et al. (2012) (P6 Based on the variance 2)

Description

The function BS8() 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

BS8(mu.link = "log", sigma.link = "log")

Value

Returns a gamlss.family object which can be used to fit a BS8 distribution in the gamlss() function.

Arguments

mu.link

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

sigma.link

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

Author

David Villegas Ceballos,

Details

The Birnbaum-Saunders distribution with parameters mu and sigma (where mu represents the true variance \(\sigma^2\) and sigma represents the shape parameter \(\alpha\)) has density given by

\(f(x|\mu,\sigma) = \frac{\sqrt{\mu}} {2\sqrt{2\pi\sigma}} \left[ \left\{ \frac{1}{2x} \sqrt{\frac{5\sigma}{\mu(\mu-1)}} \right\}^{1/2} + \left\{ \frac{1}{2x} \sqrt{\frac{5\sigma}{\mu(\mu-1)}} \right\}^{3/2} \right] \exp\left( -\frac{5}{8(\mu-1)} \left[ \frac{2x\sqrt{\mu(\mu-1)}}{\sqrt{5\sigma}} + \frac{\sqrt{5\sigma}} {2+\sqrt{\mu(\mu-1)}} -2 \right] \right) \)

for \(x>0\), \(\mu>0\) and \(\sigma>0\). In this parameterization, \(E(X) = \frac{[2\mu+3]\sqrt{\sigma}}{\sqrt{20\mu(\mu-1)}}\) and \(Var(X) = \sigma\).

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

dBS8.

Examples

# Example 1
# Generating some random values with
# known mu and sigma

set.seed(12345)
y <- rBS8(n=100, mu=2, sigma=10)

# Fitting the model using default link function
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS8,
               control=gamlss.control(n.cyc=1000))

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

# Fitting the model using a own link function

# 1. Extract the built-in "logshiftto1" link structure
if (FALSE) {
require(gamlss.dist)
logshift_link <- make.link.gamlss("logshiftto1")

# 2. Assign its components to the 'own' functions gamlss searches for
own.linkfun  <- logshift_link$linkfun
own.linkinv  <- logshift_link$linkinv
own.mu.eta   <- logshift_link$mu.eta
own.valideta <- logshift_link$valideta

mod99 <- gamlss(y~1, sigma.fo=~1, family=BS8(mu.link = "own"),
                control=gamlss.control(n.cyc=1000))

# Extracting the fitted values for mu and sigma
# using the inverse link function
own.linkinv(coef(mod99, what="mu"))
exp(coef(mod00, what="sigma"))
}

# Example 2
# Generating random values for a regression model

# A function to simulate a data set with Y ~ BS8
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.6 - 1.4 * x1)     # Aprox 2.45 with link log(mu)
  sigma <- exp(1.6 + 1.5 * x2)  # Aprox 10
  y <- rBS8(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(1234)
dat <- gendat(n=400)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS8, data=dat,
               control=gamlss.control(n.cyc=100))

summary(mod2)