Birnbaum–Saunders distribution

The Birnbaum–Saunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times. There are several alternative formulations of this distribution in the literature. It is named after Z. W. Birnbaum and S. C. Saunders.

This distribution was developed to model failures due to cracks. A material is placed under repeated cycles of stress. The j^th cycle leads to an increase in the crack by X_j amount. The sum of the X_j is assumed to be normally distributed with mean nμ and variance nσ². The probability that the crack does not exceed a critical length ω is

${\displaystyle P(X\leq \omega )=\Phi \left({\frac {\omega -n\mu }{\sigma {\sqrt {n}}}}\right)}$

where Φ() is the cdf of normal distribution.

If T is the number of cycles to failure then the cumulative distribution function (cdf) of T is

${\displaystyle P(T\leq t)=1-\Phi \left({\frac {\omega -t\mu }{\sigma {\sqrt {t}}}}\right)=\Phi \left({\frac {t\mu -\omega }{\sigma {\sqrt {t}}}}\right)=\Phi \left({\frac {\mu {\sqrt {t}}}{\sigma }}-{\frac {\omega }{\sigma {\sqrt {t}}}}\right)=\Phi \left({\frac {\sqrt {\mu \omega }}{\sigma }}\left[\left({\frac {t}{\omega /\mu }}\right)^{0.5}-\left({\frac {\omega /\mu }{t}}\right)^{0.5}\right]\right)}$

The more usual form of this distribution is:

${\displaystyle F(x;\alpha ,\beta )=\Phi \left({\frac {1}{\alpha }}\left[\left({\frac {x}{\beta }}\right)^{0.5}-\left({\frac {\beta }{x}}\right)^{0.5}\right]\right)}$

Here α is the shape parameter and β is the scale parameter.

The Birnbaum–Saunders distribution is unimodal with a median of β.

The mean (μ), variance (σ²), skewness (γ) and kurtosis (κ) are as follows:

${\displaystyle \mu =\beta \left(1+{\frac {\alpha ^{2}}{2}}\right)}$

${\displaystyle \sigma ^{2}=(\alpha \beta )^{2}\left(1+{\frac {5\alpha ^{2}}{4}}\right)}$

${\displaystyle \gamma ={\frac {4\alpha (11\alpha ^{2}+6)}{(5\alpha ^{2}+4)^{\frac {3}{2}}}}}$

${\displaystyle \kappa =3+{\frac {6\alpha ^{2}(93\alpha ^{2}+40)}{(5\alpha ^{2}+4)^{2}}}}$

Given a data set that is thought to be Birnbaum–Saunders distributed the parameters' values are best estimated by maximum likelihood.

If T is Birnbaum–Saunders distributed with parameters α and β then T⁻¹ is also Birnbaum-Saunders distributed with parameters α and β⁻¹.

Let T be a Birnbaum-Saunders distributed variate with parameters α and β. A useful transformation of T is

${\displaystyle X={\frac {1}{2}}\left[\left({\frac {T}{\beta }}\right)^{0.5}-\left({\frac {T}{\beta }}\right)^{-0.5}\right]}$.

Equivalently

${\displaystyle T=\beta \left(1+2X^{2}+2X(1+X^{2})^{0.5}\right)}$.

X is then distributed normally with a mean of zero and a variance of α² / 4.

The general formula for the probability density function (pdf) is

${\displaystyle f(x)={\frac {{\sqrt {\frac {x-\mu }{\beta }}}+{\sqrt {\frac {\beta }{x-\mu }}}}{2\gamma \left(x-\mu \right)}}\phi \left({\frac {{\sqrt {\frac {x-\mu }{\beta }}}-{\sqrt {\frac {\beta }{x-\mu }}}}{\gamma }}\right)\quad x>\mu ;\gamma ,\beta >0}$

where γ is the shape parameter, μ is the location parameter, β is the scale parameter, and ${\displaystyle \phi }$ is the probability density function of the standard normal distribution.

The case where μ = 0 and β = 1 is called the standard fatigue life distribution. The pdf for the standard fatigue life distribution reduces to

${\displaystyle f(x)={\frac {{\sqrt {x}}+{\sqrt {\frac {1}{x}}}}{2\gamma x}}\phi \left({\frac {{\sqrt {x}}-{\sqrt {\frac {1}{x}}}}{\gamma }}\right)\quad x>0;\gamma >0}$

Since the general form of probability functions can be expressed in terms of the standard distribution, all of the subsequent formulas are given for the standard form of the function.

The formula for the cumulative distribution function is

${\displaystyle F(x)=\Phi \left({\frac {{\sqrt {x}}-{\sqrt {\frac {1}{x}}}}{\gamma }}\right)\quad x>0;\gamma >0}$

where Φ is the cumulative distribution function of the standard normal distribution.

The formula for the quantile function is

${\displaystyle G(p)={\frac {1}{4}}\left[\gamma \Phi ^{-1}(p)+{\sqrt {4+\left(\gamma \Phi ^{-1}(p)\right)^{2}}}\right]^{2}}$

where Φ ⁻¹ is the quantile function of the standard normal distribution.

The Birnbaum-Saunders distribution can be shown to be the result of an equal mixture between an inverse Gaussian and a reciprocal inverse Gaussian distribution.

This property can be conceptualized by considering the Wiener process with drift: the first hitting time to a point will have an inverse Gaussian distribution, the last hitting time to that point will have a reciprocal inverse Gaussian distribution, and the Birnbaum-Saunders distribution would represent the probability the process is beyond the point irrespective of how many times it has or will pass the point.

• Birnbaum, Z. W.; Saunders, S. C. (1969), "A new family of life distributions", Journal of Applied Probability, 6 (2): 319–327, doi:10.2307/3212003, JSTOR 3212003, archived from the original on September 23, 2017
• Desmond, A.F. (1985), "Stochastic models of failure in random environments", Canadian Journal of Statistics, 13 (3): 171–183, doi:10.2307/3315148, JSTOR 3315148
• Johnson, N.; Kotz, S.; Balakrishnan, N. (1995), Continuous Univariate Distributions, vol. 2 (2nd ed.), New York: Wiley
• Lemonte, A. J.; Cribari-Neto, F.; Vasconcellos, K. L. P. (2007), "Improved statistical inference for the two-parameter Birnbaum–Saunders distribution", Computational Statistics and Data Analysis, 51 (9): 4656–4681, doi:10.1016/j.csda.2006.08.016
• Lemonte, A. J.; Simas, A. B.; Cribari-Neto, F. (2008), "Bootstrap-based improved estimators for the two-parameter Birnbaum–Saunders distribution", Journal of Statistical Computation and Simulation, 78: 37–49, doi:10.1080/10629360600903882
• Cordeiro, G. M.; Lemonte, A. J. (2011), "The β-Birnbaum–Saunders distribution: An improved distribution for fatigue life modeling", Computational Statistics and Data Analysis, 55 (3): 1445–1461, doi:10.1016/j.csda.2010.10.007

• Fatigue life distribution

 This article incorporates public domain material from the National Institute of Standards and Technology