bsgof (version 0.23.8)

BS: The Birnbaum-Saunders distribution

Description

Density, distribution function, quantile function and random generation for the Birnbaum-Saunders distribution with alpha (shape) and beta (scale)

Usage

dbs(x, alpha = 1, beta = 1, log = FALSE)
pbs(q, alpha = 1, beta = 1, lower.tail = TRUE, log.p = FALSE)
qbs(p, alpha = 1, beta = 1, lower.tail = TRUE, log.p = FALSE)
rbs(n, alpha = 1, beta = 1)

Value

dbs gives the density, pbs gives the distribution function, qbs gives the quantile function, and rbs generates random deviates.

Arguments

x,q

vector of quantiles.

p

vector of probabilities.

n

number of observations.

alpha

shape parameter.

beta

scale parameter.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\) otherwise, \(P[X > x]\).

Author

Chanseok Park

Details

The Birnbaum-Saunders distribution was proposed by Birnbaum and Saunders (1969) and its probability density function and cumulative distribution function are given by $$f(x) = \frac{1}{\sqrt{2\pi}} \exp\left[-\frac{1}{2\alpha^{2}} \left(\frac{x}{\beta}+\frac{\beta}{x}-2\right) \right] \frac{x^{-\frac{3}{2}} (x+\beta)}{2\alpha\sqrt{\beta}}$$ and $$F(x) = \Phi \Big[ \frac{1}{\alpha} \Big( \sqrt{\frac{x}{\beta}}-\sqrt{\frac{\beta}{x}} \Big) \Big],$$ where \(x>0\), \(\alpha>0\), and \(\beta>0\).

References

Birnbaum, Z. W. and Saunders, S. C. (1969). A new family of life distributions. J. Appl. Probab. 6(2): 637-652.

Examples

dbs(1.5,  alpha=0.5, beta=1.5)
exp( dbs(1.5, alpha=0.5, beta=1.5, log=TRUE) )

pbs(2.5,  alpha=0.5, beta=1.5)
1 - pbs(2.5, alpha=0.5,beta=1.5, lower.tail = FALSE, log.p = FALSE)
1 - exp( pbs(2.5, alpha=0.5,beta=1.5, lower.tail = FALSE, log.p = TRUE) )

qbs(0.1, alpha=0.5, beta=1.5)
qbs(0.9, alpha=0.5, beta=1.5, lower.tail = FALSE, log.p = FALSE)
qbs(log(0.1), alpha=0.5, beta=1.5, lower.tail = TRUE,  log.p = TRUE)
qbs(log(0.9), alpha=0.5, beta=1.5, lower.tail = FALSE, log.p = TRUE)

rbs(n=10, alpha=0.5, beta=1.5)