RelDists (version 1.0.2)

dBS10: The Birnbaum-Saunders distribution - Santos-Neto et al. (2012) (P8 Based on the first Tweedie)

Description

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)

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.

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.

Author

David Villegas Ceballos,

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.

See Also

BS10.

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)