BS: The Birnbaum-Saunders family

In RelDists: Estimation for some Reliability Distributions

The Birnbaum-Saunders family

Description

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

Usage

BS(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{x^{-3/2}(x+\mu)}{2\sigma\sqrt{2\pi\mu}} \exp\left(\frac{-1}{2\sigma^2}(\frac{x}{\mu}+\frac{\mu}{x}-2)\right)

for x>0, \mu>0 and \sigma>0. In this parameterization \mu is the median of X, E(X)=\mu(1+\sigma^2/2) and Var(X)=(\mu\sigma)^2(1+5\sigma^2/4). The functions proposed here corresponds to the functions created by Roquim et al. (2021) with minor modifications to obtain correct log-likelihoods and random samples.

Value

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

References

Birnbaum, Z.W. and Saunders, S.C. (1969a). A new family of life distributions. J. Appl. Prob., 6, 319–327.

Roquim, F. V., Ramires, T. G., Nakamura, L. R., Righetto, A. J., Lima, R. R., & Gomes, R. A. (2021). Building flexible regression models: including the Birnbaum-Saunders distribution in the gamlss package. Semina: Ciencias Exatas e Tecnologicas, 42(2), 163-168.

See Also

dBS

Examples

# Example 1
# Generating some random values with
# known mu and sigma
y <- rBS(n=100, mu=0.75, sigma=1.3)

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

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

set.seed(123)
dat <- gendat(n=300)

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

summary(mod2)

# Example 3
# Fatigue life (T) measures in cycles (×10-3) of n 101
# aluminum coupons (specimens) of type 6061-T6.
# Taken from Leiva et al. (2006) page 37.
# https://journal.r-project.org/articles/RN-2006-033/RN-2006-033.pdf

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)

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

# 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
# Aggregate payments by the insurer
# in thousand Skr (Swedish currency).
# Taken from Balakrishnan and Kundu (2019) page 65.
# https://onlinelibrary.wiley.com/doi/abs/10.1002/asmb.2348

y <- c(5014, 5855, 6486, 6540, 6656, 6656, 7212, 7541, 7558, 
       7797, 8546, 9345, 11762, 12478, 13624, 14451,
       14940, 14963, 15092, 16203, 16229, 16730, 18027, 
       18343, 19365, 21782, 24248, 29069, 34267, 38993)

y <- y/10000

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

# 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.