The Birnbaum-Saunders distribution - Santos-Neto et al. (2012) (P8 Based on the the first Tweedie)

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

Usage

dBS10(x, mu = 1, sigma = 0.5, log = FALSE)

pBS10(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)

qBS10(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)

rBS10(n, mu = 1, sigma = 0.5)

hBS10(x, mu, sigma)

Arguments

x, q: vector of quantiles.
mu: parameter representing \(\tau\) (mu > 0).
sigma: parameter representing \(\omega\) (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

dBS10 gives the density, pBS10 gives the distribution function, qBS10 gives the quantile function, rBS10 generates random deviates and hBS10 gives the hazard function.

Details

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

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

for \(x>0\), \(\mu>0\) and \(\sigma>0\). In this parameterization, \(E(X) = \frac{\mu^2}{2} + \frac{\mu^4}{8\sigma}\) and \(Var(X) = \frac{\mu^6}{8\sigma} + \frac{5\mu^8}{64\sigma^2}\).

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(dBS10(x, mu=0.75, sigma=5), 
      from=0.001, to=1.5,
      ylim=c(0, 6.2),
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS10(x, mu=1.15, sigma=5),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=0.75, sigma=5", 
                            "mu=1.15, sigma=5"),
       col=c("royalblue1", "tomato"), lwd=2, cex=0.6)


# Example 2
# Checking if the cumulative curves converge to 1
curve(pBS10(x, mu=0.75, sigma=5), 
      from=0.00001, to=1.5,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="F(x)")
curve(pBS10(x, mu=1.15, sigma=5),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=0.75, sigma=5", 
                               "mu=1.15, sigma=5"),
       col=c("royalblue1", "tomato"), lwd=2, cex=0.5)


# Example 3
# The quantile function
p <- seq(from=0, to=0.999, length.out=100)
plot(x=qBS10(p, mu=0.75, sigma=5), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pBS10(x, mu=0.75, sigma=5), 
      from=0, add=TRUE, col="tomato", lwd=2.5)


# Example 4
# The random function
x <- rBS10(n=10000, mu=0.75, sigma=5)
hist(x, freq=FALSE)
curve(dBS10(x, mu=0.75, sigma=5),  
      add=TRUE, col="tomato", lwd=2)


# Example 5
# The Hazard function
curve(hBS10(x, mu=0.75, sigma=5), from=0.001, to=2,
      col="tomato", ylab="Hazard function", las=1)