dBS2: The Birnbaum-Saunders distribution - Santos-Neto et al....

In RelDists: Estimation for some Reliability Distributions

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

Description

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

Usage

dBS2(x, mu = 1, sigma = 1, log = FALSE)

pBS2(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qBS2(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rBS2(n, mu = 1, sigma = 1)

hBS2(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) = \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)=(\mu\sigma)^2(1+5\sigma^2/4). The functions proposed here corresponds to the parameterization proposed by Santos-Neto et al. (2014).

Value

dBS2 gives the density, pBS2 gives the distribution function, qBS2 gives the quantile function, rBS2 generates random deviates and hBS2 gives the hazard 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

BS2.

Examples

#Example 1
#Plotting the mass function for different parameter values
curve(dBS2(x, mu=1.0, sigma=100), 
      from=0.001, to=5,
      ylim=c(0, 3), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS2(x, mu=1.5, sigma=100),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dBS2(x, mu=2.0, sigma=100),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=1.0, sigma=100", 
                            "mu=1.5, sigma=100",
                            "mu=2.0, sigma=100"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)


curve(dBS2(x, mu=1, sigma=2), 
      from=0.001, to=2,
      ylim=c(0, 1.1), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS2(x, mu=1, sigma=5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dBS2(x, mu=1, sigma=10),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=1, sigma=2", 
                            "mu=1, sigma=5",
                            "mu=1, sigma=10"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)


# Example 2
# Checking if the cumulative curves converge to 1
curve(pBS2(x, mu=0.5, sigma=0.5), 
      from=0.001, to=15,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="f(x)")
curve(pBS2(x, mu=1, sigma=0.5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(pBS2(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)

# Example 3
# The quantile function
p <- seq(from=0, to=0.999, length.out=100)
plot(x=qBS2(p, mu=2.3, sigma=1.7), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pBS2(x, mu=2.3, sigma=1.7), 
      from=0, add=TRUE, col="tomato", lwd=2.5)

# Example 4
# The random function
x <- rBS2(n=10000, mu=2.5, sigma=100)
hist(x, freq=FALSE)
curve(dBS2(x, mu=2.5, sigma=100), from=0, to=10, 
      add=TRUE, col="tomato", lwd=2)

# Example 5
# The Hazard function
curve(hBS2(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.