BS2: The Birnbaum-Saunders family - Santos-Neto et al. (2014)

In RelDists: Estimation for some Reliability Distributions

The Birnbaum-Saunders family - Santos-Neto et al. (2014)

Description

The function BS2() defines The Birnbaum-Saunders, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS2(mu.link = "log", sigma.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.

Details

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

f(x|\mu,\sigma) = \frac{\exp(\sigma/2)\sqrt{\sigma+1}}{4\sqrt{\pi\mu}x^{3/2}} \left[ x + \frac{\mu\sigma}{\sigma+1} \right] \exp\left( \frac{-\sigma}{4} \left(\frac{x(\sigma+1)}{\mu\sigma}+\frac{\mu\sigma}{x(\sigma+1)} \right) \right)

for x>0, \mu>0 and \sigma>0. In this parameterization E(X)=\mu and Var(X)=\frac{\mu^2(2\sigma+5)}{(\sigma+1)^2}.

Value

Returns a gamlss.family object which can be used to fit a BS2 distribution in the gamlss() function.

References

Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., & Barros, M. (2014). A reparameterized Birnbaum–Saunders distribution and its moments, estimation and applications. REVSTAT-Statistical Journal, 12(3), 247-272.

See Also

dBS2.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
y <- rBS2(n=50, mu=5, sigma=3)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS2)

# 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 ~ BS2
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.45 - 3 * x1)
  sigma <- exp(2 - 1.5 * x2)
  y <- rBS2(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(123)
dat <- gendat(n=100)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS2, data=dat)

summary(mod2)

# Example 3
# Household expenditures for food in the United States (US) expressed
# in thousands of US dollars (M$)
# Santos-Neto et al. (2014) page 266.

y <- c(15.998, 16.652, 21.741, 7.431, 10.481, 13.548, 23.256, 17.976, 
       14.161, 8.825, 14.184, 19.604, 13.728, 21.141, 17.446, 9.629, 
       14.005, 9.160, 18.831, 7.641, 13.882, 9.670, 21.604, 10.866, 
       28.980, 10.882, 18.561, 11.629, 18.067, 14.539, 19.192, 25.918, 
       28.833, 15.869, 14.910, 9.550, 23.066, 14.751)

mod3 <- gamlss(y~1, sigma.fo=~1, family=BS2)

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod3, what="mu"))
exp(coef(mod3, what="sigma"))

# Example 4
# lifetimes of 6061-T6 aluminum coupons expressed in cycles (×10-3)
# at a maximum stress level of 3.1 psi (×104), until the failure to occur.
# Santos-Neto et al. (2014) page 267.

y <- c(70, 90, 96, 97, 99, 100, 103, 104, 104, 105, 107, 108, 108, 108, 109,
       109, 112, 112, 113, 114, 114, 114, 116, 119, 120, 120, 120, 121, 121, 
       123, 124, 124, 124, 124, 124, 128, 128, 129, 129, 130, 130, 130, 131, 
       131, 131, 131, 131, 132, 132, 132, 133, 134, 134, 134, 134, 134, 136, 
       136, 137, 138, 138, 138, 139, 139, 141, 141, 142, 142, 142, 142, 142, 
       142, 144, 144, 145, 146, 148, 148, 149, 151, 151, 152, 155, 156, 157, 
       157, 157, 157, 158, 159, 162, 163, 163, 164, 166, 166, 168, 170, 174, 
       196, 212)

mod4 <- gamlss(y~1, sigma.fo=~1, family=BS2)

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod4, what="mu"))
exp(coef(mod4, what="sigma"))

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