The Birnbaum-Saunders distribution - Santos-Neto et al. (2012) (P4 Based on the mean)

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

Usage

dBS6(x, mu = 1, sigma = 0.5, log = FALSE)

pBS6(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)

qBS6(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)

rBS6(n, mu = 1, sigma = 0.5)

hBS6(x, mu, sigma)

Arguments

x, q: vector of quantiles.
mu: parameter representing the mean (mu > 0).
sigma: parameter representing the shape \(\alpha\) (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

dBS6 gives the density, pBS6 gives the distribution function, qBS6 gives the quantile function, rBS6 generates random deviates and hBS6 gives the hazard function.

Details

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

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

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

References

Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., & Ahmed, S. E. (2012). On new parameterizations of the Birnbaum-Saunders distribution. Pakistan Journal of Statistics, 28(1), 1-26.

Author

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

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dBS6(x, mu=2, sigma=0.1), 
      from=0.001, to=3,
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS6(x, mu=2, sigma=0.75),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=2, sigma=0.1", 
                            "mu=2, sigma=0.75"),
       col=c("royalblue1", "tomato"), lwd=2, cex=0.6)


# Example 2
# Checking if the cumulative curves converge to 1
curve(pBS6(x, mu=2, sigma=0.1), 
      from=0.00001, to=6,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="F(x)")
curve(pBS6(x, mu=2, sigma=0.75),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=2, sigma=0.1", 
                               "mu=2, sigma=0.75"),
       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=qBS6(p, mu=1, sigma=2), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pBS6(x, mu=1, sigma=2), 
      from=0, add=TRUE, col="tomato", lwd=2.5)


# Example 4
# The random function
x <- rBS6(n=10000, mu=2, sigma=0.1)
hist(x, freq=FALSE)
curve(dBS6(x, mu=2, sigma=0.1),  
      add=TRUE, col="tomato", lwd=2)


# Example 5
# The Hazard function
curve(hBS6(x, mu=2, sigma=0.1), from=0.001, to=6,
      col="tomato", ylab="Hazard function", las=1)