The Birnbaum-Saunders distribution - Second parameterization

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

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.

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dBS3(x, mu=2, sigma=0.2), 
      from=0.001, to=10,
      ylim=c(0, 0.4), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS3(x, mu=2, sigma=0.4),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=2, sigma=0.2", 
                            "mu=2, sigma=0.4"),
       col=c("royalblue1", "tomato"), lwd=2, cex=0.6)


# Example 2
# Checking if the cumulative curves converge to 1
curve(pBS3(x, mu=2, sigma=0.2), 
      from=0.00001, to=10,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="F(x)")
curve(pBS3(x, mu=2, sigma=0.4),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=2, sigma=0.2", 
                               "mu=2, sigma=0.4"),
       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=qBS3(p, mu=2, sigma=0.2), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pBS3(x, mu=2, sigma=0.2), 
      from=0, add=TRUE, col="tomato", lwd=2.5)


# Example 4
# The random function
x <- rBS3(n=10000, mu=2, sigma=0.2)
hist(x, freq=FALSE)
curve(dBS3(x, mu=2, sigma=0.2), from=0, to=10, 
      add=TRUE, col="tomato", lwd=2)


# Example 5
# The Hazard function
curve(hBS3(x, mu=2, sigma=0.2), from=0.001, to=4,
      col="tomato", ylab="Hazard function", las=1)