The Birnbaum-Saunders distribution - Ahmed et al. (2008)

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

Usage

dBS4(x, mu = 1, sigma = 0.5, log = FALSE)

pBS4(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)

qBS4(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)

rBS4(n, mu = 1, sigma = 0.5)

hBS4(x, mu, sigma)

Arguments

x, q: vector of quantiles.
mu: parameter (mu > 0).
sigma: parameter (sigma > 0).
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.

Value

dBS4 gives the density, pBS4 gives the distribution function, qBS4 gives the quantile function, rBS4 generates random deviates and hBS4 gives the hazard function.

Details

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

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

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

References

Ahmed, S. E., Budsaba, K., Lisawadi, S., & Volodin, A. (2008). Parametric estimation for the Birnbaum-Saunders lifetime distribution based on a new parametrization. Thailand Statistician, 6(2), 213-240.

Author

David Villegas Ceballos, david.villegas1@udea.edu.co

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dBS4(x, mu=2, sigma=30), 
      from=0.001, to=40,
      ylim=c(0, 0.20), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS4(x, mu=1, sigma=20),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=2, sigma=30", 
                            "mu=1, sigma=20"),
       col=c("royalblue1", "tomato"), lwd=2, cex=0.6)


# Example 2
# Checking if the cumulative curves converge to 1
curve(pBS4(x, mu=2, sigma=30), 
      from=0.00001, to=40,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="F(x)")
curve(pBS4(x, mu=1, sigma=20),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=2, sigma=30", 
                               "mu=1, sigma=20"),
       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=qBS4(p, mu=2, sigma=30), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pBS4(x, mu=2, sigma=30), 
      from=0, add=TRUE, col="tomato", lwd=2.5)


# Example 4
# The random function
x <- rBS4(n=10000, mu=2, sigma=30)
hist(x, freq=FALSE)
curve(dBS4(x, mu=2, sigma=30), from=0, to=30, 
      add=TRUE, col="tomato", lwd=2)


# Example 5
# The Hazard function
curve(hBS4(x, mu=2, sigma=30), from=0.001, to=40,
      col="tomato", ylab="Hazard function", las=1)