RelDists (version 1.0.2)

dBS4: The Birnbaum-Saunders distribution - Ahmed et al. (2008)

Description

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

Usage

dBS4(x, mu = 1, sigma = 0.5, log = FALSE)
pBS4(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
qBS4(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
rBS4(n, mu = 1, sigma = 0.5)
hBS4(x, mu, sigma)

Value

dBS4 gives the density, pBS4 gives the distribution function, qBS4 gives the quantile function, rBS4

generates random deviates and hBS4 gives the hazard function.

Arguments

x, q

vector of quantiles.

mu

parameter (mu > 0).

sigma

parameter (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

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

for \(x>0\), \(\mu>0\) and \(\sigma>0\). In this parameterization \(E(X) = \frac{\sigma \mu + 1/2}{\mu^2}\) and \(Var(X) = \frac{\sigma \mu + 5/4}{\mu^4}\).

References

Ahmed, S. E., Budsaba, K., Lisawadi, S., & Volodin, A. (2008). Parametric estimation for the Birnbaum-Saunders lifetime distribution based on a new parametrization. Thailand Statistician, 6(2), 213-240.

See Also

BS4.

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dBS4(x, mu=2, sigma=30), 
      from=0.001, to=40,
      ylim=c(0, 0.20), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS4(x, mu=1, sigma=20),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=2, sigma=30", 
                            "mu=1, sigma=20"),
       col=c("royalblue1", "tomato"), lwd=2, cex=0.6)

# Example 2
# Checking if the cumulative curves converge to 1
curve(pBS4(x, mu=2, sigma=30), 
      from=0.00001, to=40,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="F(x)")
curve(pBS4(x, mu=1, sigma=20),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=2, sigma=30", 
                               "mu=1, sigma=20"),
       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=qBS4(p, mu=2, sigma=30), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pBS4(x, mu=2, sigma=30), 
      from=0, add=TRUE, col="tomato", lwd=2.5)

# Example 4
# The random function
x <- rBS4(n=10000, mu=2, sigma=30)
hist(x, freq=FALSE)
curve(dBS4(x, mu=2, sigma=30), from=0, to=30, 
      add=TRUE, col="tomato", lwd=2)

# Example 5
# The Hazard function
curve(hBS4(x, mu=2, sigma=30), from=0.001, to=40,
      col="tomato", ylab="Hazard function", las=1)