BS {bsgof}

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)

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].

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.

Value

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

Author(s)

Chanseok Park

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)