The Birnbaum-Saunders distribution - Bourguignon & Gallardo (2022)

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

Usage

dBS3(x, mu = 1, sigma = 0.5, log = FALSE)

pBS3(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)

qBS3(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)

rBS3(n, mu = 1, sigma = 0.5)

hBS3(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

dBS3 gives the density, pBS3 gives the distribution function, qBS3 gives the quantile function, rBS3 generates random deviates and hBS3 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

Bourguignon, M., & Gallardo, D. I. (2022). A new look at the Birnbaum–Saunders regression model. Applied Stochastic Models in Business and Industry, 38(6), 935-951.

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)