BirnbaumSaunders function

Birnbaum-Saunders (fatigue life) distribution

Birnbaum-Saunders (fatigue life) distribution

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

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≤x]P[X \le x]

    otherwise, P[X>x]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)=(x−μβ+βx−μ2α(x−μ))ϕ(1α(x−μβ−βx−μ))f(x)=((sqrt((x−μ)/β)+sqrt(β/(x−μ)))/(2∗α∗(x−μ)))∗ϕ((sqrt((x−μ)/β)−sqrt(β/(x−μ)))/α) 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)f(x) = ((sqrt((x-\mu)/\beta) + sqrt(\beta/(x-\mu)))/(2*\alpha*(x-\mu))) *\phi((sqrt((x-\mu)/\beta) - sqrt(\beta/(x-\mu)))/\alpha)

Cumulative distribution function

F(x)=Φ(1α(x−μβ−βx−μ))F(x)=Φ(((sqrt((x−μ)/β)−sqrt(β/(x−μ)))/α) F(x) = \Phi \left(\frac{1}{\alpha}\left( \sqrt{\frac{x-\mu}{\beta}} -\sqrt{\frac{\beta}{x-\mu}} \right) \right)F(x) = \Phi(((sqrt((x-\mu)/\beta) - sqrt(\beta/(x-\mu)))/\alpha)

Quantile function

F−1(p)=[α2Φ−1(p)+(α2Φ−1(p))2+1]2β+\muF−1(p)=(α/2∗Φ−1(p)+sqrt((α/2∗Φ−1(p))2+1)2∗β+μ F^{-1}(p) = \left[\frac{\alpha}{2} \Phi^{-1}(p) +\sqrt{\left(\frac{\alpha}{2} \Phi^{-1}(p)\right)^{2} + 1}\right]^{2} \beta + \muF^-1(p) = (\alpha/2 * \Phi^-1(p) +sqrt((\alpha/2 * \Phi^-1(p))^2 + 1)^2 * \beta + \mu

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)

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.

extraDistr packageRead PDF manual
  • Maintainer: Tymoteusz Wolodzko
  • License: GPL-2
  • Last published: 2023-11-30

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