dBS: The Birnbaum-Saunders distribution

In RelDists: Estimation for some Reliability Distributions

The Birnbaum-Saunders distribution

Description

Density, distribution function, quantile function, random generation and hazard function for the Birnbaum-Saunders distribution with parameters mu and sigma.

Usage

dBS(x, mu = 1, sigma = 1, log = FALSE)

pBS(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qBS(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rBS(n, mu = 1, sigma = 1)

hBS(x, mu, sigma)

Arguments

x, q	vector of quantiles.
mu	parameter.
sigma	parameter.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
p	vector of probabilities.
n	number of observations.

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

dBS gives the density, pBS gives the distribution function, qBS gives the quantile function, rBS generates random deviates and hBS gives the hazard 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

BS.

Examples

#Example 1
#Plotting the mass function for different parameter values
curve(dBS(x, mu=0.5, sigma=0.5), 
      from=0.001, to=5,
      ylim=c(0, 2), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS(x, mu=1, sigma=0.5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dBS(x, mu=1.5, sigma=0.5),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=0.5, sigma=0.5", 
                            "mu=1.0, sigma=0.5",
                            "mu=1.5, sigma=0.5"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)


curve(dBS(x, mu=0.5, sigma=1), 
      from=0.001, to=5,
      ylim=c(0, 1.5), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS(x, mu=1, sigma=1),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dBS(x, mu=1.5, sigma=1),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=0.5, sigma=1", 
                            "mu=1.0, sigma=1",
                            "mu=1.5, sigma=1"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)


curve(dBS(x, mu=0.5, sigma=1.5), 
      from=0.001, to=8,
      ylim=c(0, 2), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS(x, mu=1, sigma=1.5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dBS(x, mu=1.5, sigma=1.5),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=0.5, sigma=1.5", 
                            "mu=1.0, sigma=1.5",
                            "mu=1.5, sigma=1.5"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)

# Example 2
# Checking if the cumulative curves converge to 1
curve(pBS(x, mu=0.5, sigma=0.5), 
      from=0.001, to=5,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="f(x)")
curve(pBS(x, mu=1, sigma=0.5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(pBS(x, mu=1.5, sigma=0.5),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=0.5, sigma=0.5", 
                               "mu=1.0, sigma=0.5",
                               "mu=1.5, sigma=0.5"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.5)
curve(pBS(x, mu=0.5, sigma=0.5, lower.tail=FALSE), 
      from=0.001, to=5,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="f(x)")
curve(pBS(x, mu=1, sigma=0.5, lower.tail=FALSE),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(pBS(x, mu=1.5, sigma=0.5, lower.tail=FALSE),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=0.5, sigma=0.5", 
                            "mu=1.0, sigma=0.5",
                            "mu=1.5, sigma=0.5"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.5)

#example 3
## The quantile function
p <- seq(from=0, to=0.999, length.out=100)
plot(x=qBS(p, mu=2.3, sigma=1.7), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pBS(x, mu=2.3, sigma=1.7), 
      from=0, add=TRUE, col="tomato", lwd=2.5)

#some values
p <- c(0.25, 0.5, 0.75)
quantile <- qBS(p=p, mu=2.3, sigma=1.7) 
for(i in quantile){
  print(integrate(dBS, lower=0, upper=i, mu=2.3, sigma=1.7))
}

#example 4
## The random function
x <- rBS(n=10000, mu=20, sigma=0.5)
hist(x, freq=FALSE)
curve(dBS(x, mu=20, sigma=0.5), from=0, to=100, 
      add=TRUE, col="tomato", lwd=2)

#example 5
## The Hazard function
curve(hBS(x, mu=20, sigma=0.5), from=0.001, to=100,
      col="tomato", ylab="Hazard function", las=1)

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