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

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.

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.

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).

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.

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)