extraDistr (version 1.9.1)

BirnbaumSaunders: Birnbaum-Saunders (fatigue life) distribution

Description

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

Usage

dfatigue(x, alpha, beta = 1, mu = 0, log = FALSE)
pfatigue(q, alpha, beta = 1, mu = 0, lower.tail = TRUE, log.p = FALSE)
qfatigue(p, alpha, beta = 1, mu = 0, lower.tail = TRUE, log.p = FALSE)
rfatigue(n, alpha, beta = 1, mu = 0)

Arguments

x, q

vector of quantiles.

alpha, beta, mu

shape, scale and location parameters. Scale and shape must be positive.

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, \(P[X > x]\).

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function $$ f(x) = \left (\frac{\sqrt{\frac{x-\mu} {\beta}} + \sqrt{\frac{\beta} {x-\mu}}} {2\alpha (x-\mu)} \right) \phi \left( \frac{1}{\alpha}\left( \sqrt{\frac{x-\mu}{\beta}} - \sqrt{\frac{\beta}{x-\mu}} \right) \right) $$

Cumulative distribution function $$ F(x) = \Phi \left(\frac{1}{\alpha}\left( \sqrt{\frac{x-\mu}{\beta}} - \sqrt{\frac{\beta}{x-\mu}} \right) \right) $$

Quantile function $$ F^{-1}(p) = \left[\frac{\alpha}{2} \Phi^{-1}(p) + \sqrt{\left(\frac{\alpha}{2} \Phi^{-1}(p)\right)^{2} + 1}\right]^{2} \beta + \mu $$

References

Birnbaum, Z. W. and Saunders, S. C. (1969). A new family of life distributions. Journal of Applied Probability, 6(2), 637-652.

Desmond, A. (1985) Stochastic models of failure in random environments. Canadian Journal of Statistics, 13, 171-183.

Vilca-Labra, F., and Leiva-Sanchez, V. (2006). A new fatigue life model based on the family of skew-elliptical distributions. Communications in Statistics-Theory and Methods, 35(2), 229-244.

Leiva, V., Sanhueza, A., Sen, P. K., and Paula, G. A. (2008). Random number generators for the generalized Birnbaum-Saunders distribution. Journal of Statistical Computation and Simulation, 78(11), 1105-1118.

Examples

x <- rfatigue(1e5, .5, 2, 5)
hist(x, 100, freq = FALSE)
curve(dfatigue(x, .5, 2, 5), 2, 20, col = "red", add = TRUE)
hist(pfatigue(x, .5, 2, 5))
plot(ecdf(x))
curve(pfatigue(x, .5, 2, 5), 2, 20, col = "red", lwd = 2, add = TRUE)