bisa: Birnbaum-Saunders distribution

In RTMBdist: Distributions Compatible with Automatic Differentiation by 'RTMB'

Birnbaum-Saunders distribution

Description

Density, distribution function, quantile function, and random generation for the Birnbaum-Saunders (fatigue life) distribution.

Usage

dbisa(x, scale = 1, shape, log = FALSE)

pbisa(q, scale = 1, shape, lower.tail = TRUE, log.p = FALSE)

qbisa(p, scale = 1, shape, lower.tail = TRUE, log.p = FALSE)

rbisa(n, scale = 1, shape)

Arguments

x, q	vector of quantiles.
scale	scale parameter \beta, the median, must be positive.
shape	shape parameter \alpha, must be positive.
log, log.p	logical; if TRUE, probabilities/ densities p are returned as \log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X \le x], otherwise P[X > x].
p	vector of probabilities.
n	number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

The Birnbaum-Saunders distribution with scale \beta and shape \alpha is the distribution of X for which

\xi(X) = \frac{1}{\alpha}\left(\sqrt{X/\beta} - \sqrt{\beta/X}\right) = \frac{2}{\alpha}\sinh\left(\tfrac{1}{2}\log(X/\beta)\right)

is standard normal, so that F(x) = \Phi(\xi(x)) and

f(x) = \phi(\xi(x))\, \frac{x + \beta}{2\alpha\sqrt{\beta}\, x^{3/2}}, \quad x > 0.

The scale \beta is the median, and E(X) = \beta(1 + \alpha^2/2). The quantile function is \beta \exp(2\,\mathrm{asinh}(\alpha z / 2)) with z = \Phi^{-1}(p).

Value

dbisa gives the density, pbisa gives the distribution function, qbisa gives the quantile function, and rbisa generates random deviates.

References

Birnbaum, Z. W. and Saunders, S. C. (1969) A new family of life distributions. Journal of Applied Probability, 6, 319-327.

See Also

invgauss, pgweibull, gompertz

Examples

set.seed(1)
x <- rbisa(5, scale = 2, shape = 0.5)
d <- dbisa(x, 2, 0.5)
p <- pbisa(x, 2, 0.5)
q <- qbisa(p, 2, 0.5)

RTMBdist documentation built on Oct. 31, 2026, 5:07 p.m.