BS3: The Birnbaum-Saunders family - Bourguignon & Gallardo (2022)

In RelDists: Estimation for some Reliability Distributions

The Birnbaum-Saunders family - Bourguignon & Gallardo (2022)

Description

The function BS3() defines The Birnbaum-Saunders, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS3(mu.link = "log", sigma.link = "logit")

Arguments

mu.link	defines the mu.link, with "log" link as the default for the mu parameter.
sigma.link	defines the sigma.link, with "logit" link as the default for the sigma.

Details

The Birnbaum-Saunders with parameters mu and sigma has density given by

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

for x>0, \mu>0 and 0<\sigma<1. In this parameterization Mode(X)=\mu and Var(X)=(\mu\sigma)^2(1+5\sigma^2/4).

Value

Returns a gamlss.family object which can be used to fit a BS3 distribution in the gamlss() function.

References

Bourguignon, M., & Gallardo, D. I. (2022). A new look at the Birnbaum–Saunders regression model. Applied Stochastic Models in Business and Industry, 38(6), 935-951.

See Also

dBS3.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
y <- rBS3(n=50, mu=2, sigma=0.2)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS3)

# 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 ~ BS3
## Not run: 
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.45 - 3 * x1)
  inv_logit <- function(x) 1 / (1 + exp(-x))
  sigma <- inv_logit(2 - 1.5 * x2)
  y <- rBS3(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(1234)
dat <- gendat(n=100)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS3, data=dat,
               control=gamlss.control(n.cyc=100))

summary(mod2)

## End(Not run)

# Example 3
# The response variable is the ratio between the average
# rent per acre planted with alfalfa and the corresponding 
# average rent for other agricultural uses. The density of
# dairy cows (X2, number per square mile) is the explanatory variable. 
library(alr4)
data("landrent")

landrent$ratio <- landrent$Y / landrent$X1

with(landrent, plot(x=X2, y=ratio))

mod3 <- gamlss(ratio~X2, sigma.fo=~X2, 
               data=landrent, family=BS3)

summary(mod3)
logLik(mod3)

RelDists documentation built on Feb. 24, 2026, 5:09 p.m.