RelDists (version 1.0.2)

dBS6: The Birnbaum-Saunders distribution - Santos-Neto et al. (2012) (P4 Based on the mean)

Description

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

Usage

dBS6(x, mu = 1, sigma = 0.5, log = FALSE)
pBS6(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
qBS6(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
rBS6(n, mu = 1, sigma = 0.5)
hBS6(x, mu, sigma)

Value

dBS6 gives the density, pBS6 gives the distribution function, qBS6 gives the quantile function, rBS6

generates random deviates and hBS6 gives the hazard function.

Arguments

x, q

vector of quantiles.

mu

parameter representing the mean (mu > 0).

sigma

parameter representing the shape \(\alpha\) (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

\(f(x|\mu,\sigma) = \frac{\exp(1/\sigma^2)\sqrt{2+\sigma^2}}{4\sigma\sqrt{\pi\mu}x^{3/2}} \left[ x + \frac{2\mu}{2+\sigma^2} \right] \exp\left( -\frac{1}{2\sigma^2} \left[ \frac{\{2+\sigma^2\}x}{2\mu} + \frac{2\mu}{\{2+\sigma^2\}x} \right] \right)\)

for \(x>0\), \(\mu>0\) and \(\sigma>0\). In this parameterization, \(E(X) = \mu\) and \(Var(X) = [\mu\sigma]^2 \left[ \frac{4+5\sigma^2}{(2+\sigma^2)^2} \right]\).

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

BS6.

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dBS6(x, mu=2, sigma=0.1), 
      from=0.001, to=3,
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS6(x, mu=2, sigma=0.75),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=2, sigma=0.1", 
                            "mu=2, sigma=0.75"),
       col=c("royalblue1", "tomato"), lwd=2, cex=0.6)

# Example 2
# Checking if the cumulative curves converge to 1
curve(pBS6(x, mu=2, sigma=0.1), 
      from=0.00001, to=6,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="F(x)")
curve(pBS6(x, mu=2, sigma=0.75),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=2, sigma=0.1", 
                               "mu=2, sigma=0.75"),
       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=qBS6(p, mu=1, sigma=2), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pBS6(x, mu=1, sigma=2), 
      from=0, add=TRUE, col="tomato", lwd=2.5)

# Example 4
# The random function
x <- rBS6(n=10000, mu=2, sigma=0.1)
hist(x, freq=FALSE)
curve(dBS6(x, mu=2, sigma=0.1),  
      add=TRUE, col="tomato", lwd=2)

# Example 5
# The Hazard function
curve(hBS6(x, mu=2, sigma=0.1), from=0.001, to=6,
      col="tomato", ylab="Hazard function", las=1)