RelDists (version 1.0.2)

dBS8: The Birnbaum-Saunders distribution - Santos-Neto et al. (2012) (P6 Based on the variance 2)

Description

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

Usage

dBS8(x, mu = 0.5, sigma = 10, log = FALSE)
pBS8(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
qBS8(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
rBS8(n, mu = 1, sigma = 0.5)
hBS8(x, mu, sigma)

Value

dBS8 gives the density, pBS8 gives the distribution function, qBS8 gives the quantile function, rBS8

generates random deviates and hBS8 gives the hazard function.

Arguments

x, q

vector of quantiles.

mu

parameter representing the shape (mu > 0).

sigma

parameter representing the variance (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.

Author

David Villegas Ceballos,

Details

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

\(f(x|\mu,\sigma) = \frac{\sqrt{\mu}} {2\sqrt{2\pi\sigma}} \left[ \left\{ \frac{1}{2x} \sqrt{\frac{5\sigma}{\mu(\mu-1)}} \right\}^{1/2} + \left\{ \frac{1}{2x} \sqrt{\frac{5\sigma}{\mu(\mu-1)}} \right\}^{3/2} \right] \exp\left( -\frac{5}{8(\mu-1)} \left[ \frac{2x\sqrt{\mu(\mu-1)}}{\sqrt{5\sigma}} + \frac{\sqrt{5\sigma}} {2+\sqrt{\mu(\mu-1)}} -2 \right] \right) \)

for \(x>0\), \(\mu>0\) and \(\sigma>0\). In this parameterization, \(E(X) = \frac{[2\mu+3]\sqrt{\sigma}}{\sqrt{20\mu(\mu-1)}}\) and \(Var(X) = \sigma\).

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.

See Also

BS8.

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dBS8(x, mu=1.05, sigma=10), 
      from=0.001, to=25,
      ylim=c(0, 0.25),
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS8(x, mu=1.5, sigma=10),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=1.05, sigma=10", 
                            "mu=1.5, sigma=10"),
       col=c("royalblue1", "tomato"), lwd=2, cex=0.6)

# Example 2
# Checking if the cumulative curves converge to 1
curve(pBS8(x, mu=1.5, sigma=10), 
      from=0.00001, to=30,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="F(x)")
curve(pBS8(x, mu=2.5, sigma=10),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=1.5, sigma=10", 
                               "mu=2.5, sigma=10"),
       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=qBS8(p, mu=1.5, sigma=10), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pBS8(x, mu=1.5, sigma=10), 
      from=0, add=TRUE, col="tomato", lwd=2.5)

# Example 4
# The random function
x <- rBS8(n=10000, mu=1.5, sigma=10)
hist(x, freq=FALSE)
curve(dBS8(x, mu=1.5, sigma=10),  
      add=TRUE, col="tomato", lwd=2)

# Example 5
# The Hazard function
curve(hBS8(x, mu=1.5, sigma=10), from=0.001, to=60,
      col="tomato", ylab="Hazard function", las=1)