Birnbaum-Saunders distribution

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.

Value

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

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 $z = \Phi^{-1}(p)$.

References

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

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)