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The '''Birnbaum–Saunders distribution''', also known as the '''fatigue life distribution''', is a [[probability distribution]] used extensively in [[reliability (statistics)|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.
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==Theory==
 
This distribution was developed to model failures due to cracks. A material is placed under repeated cycles of stress. The ''j''<sup>th</sup> cycle leads to an increase in the crack by ''X''<sub>j</sub> amount. The sum of the ''X''<sub>j</sub> is assumed to be normally distributed with mean ''nμ'' and variance ''nσ''<sup>2</sup>. The probability that the crack does not exceed a critical length ''ω'' is
 
<math> P( X > \omega ) = \Phi( \frac{ \omega - n \mu }{ \sigma \sqrt{ n } } )</math>
 
where ''Φ''() is the [[normal distribution]].
 
If T is the number of cycles to failure then the cumulative distribution function of T is  
 
<math>
P( T \le t ) = 1 - \Phi( \frac{ \omega - t \mu }{ \sigma \sqrt{ t } } )
= \Phi( \frac{ t \mu - \omega }{ \sigma \sqrt{ t } } )
= \Phi( \frac{ \mu \sqrt{ t } }{ \sigma } - \frac{ \omega }{ \sigma \sqrt{t} } )
= \Phi\left( \frac{ \sqrt{ \mu \omega } }{ \sigma } \left[ ( \frac{ t }{ \omega / \mu } )^{ 0.5 }  - ( \frac{ \omega / \mu }{ t } )^{ 0.5 } \right] \right)
</math>
 
The more usual form of this distribution is:
 
<math> F( x; \alpha, \beta ) = \Phi\left( \frac{ 1 }{ \alpha } \left[ ( \frac{ x }{ \beta } )^{0.5}  - ( \frac{ \beta }{ x } )^{0.5} \right] \right) </math>
 
Here ''α'' is the shape parameter and ''β'' is the location parameter.
 
==Properties==
 
The Birnbaum–Saunders distribution is unimodal with a median of ''β''.
 
The mean (''μ''), variance (σ<sup>2</sup>), skewness (''γ'') and kurtosis (''κ'') are as follows:
 
<math> \mu = \beta( 1 + \frac{ \alpha^2 }{ 2 } )</math>
 
<math> \sigma^2 = ( \alpha \beta )^2 ( 1 + \frac{ 5 \alpha^2 }{ 4 } )</math>
 
<math> \gamma = \frac{ 16 \alpha^2 ( 11 \alpha^2 + 6 ) }{ ( 5 \alpha^2 + 4 )^3 }</math>
 
<math> \kappa = 3 + \frac{ 6 \alpha^2 ( 93 \alpha^2 + 41 ) }{ ( 5 \alpha^2 + 4 )^2 }</math>
 
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''<sup>−1</sup> is also Birnbaum-Saunders distributed with parameters ''α'' and ''β''<sup>−1</sup>.
 
===Transformation===
 
Let ''T'' be a Birnbaum-Saunders distributed variate with parameters ''α'' and ''β''. A useful transformation of ''T'' is
 
<math> X = \frac{ 1 }{ 2 } [ ( \frac{ T }{ \beta } )^{ 0.5 } - ( \frac{ T }{ \beta } )^{ -0.5 } ]</math>  
 
Equivalently
 
<math> T = \beta( 1 + 2X^2 + 2X( 1 + X^2 )^{ 0.5 } ) </math>
 
''X'' is then distributed normally with a mean of zero and a variance of ''α''<sup>2</sup> / 4.
 
==Probability density function==
The general formula for the [[probability density function]] (pdf) is
:<math>
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
</math>
where γ is the [[shape parameter]], μ is the [[location parameter]], β is the [[scale parameter]], and <math>\phi \,</math> is the probability density function of the [[standard normal distribution]].
 
==Standard fatigue life 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
:<math>
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
</math>
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.
 
==Cumulative distribution function==
The formula for the [[cumulative distribution function]] is
:<math>
F(x) = \Phi\left(\frac{\sqrt{x} - \sqrt{\frac{1}{x}}}{\gamma}\right)\quad x > 0; \gamma > 0
</math>
 
where Φ is the cumulative distribution function of the standard normal distribution.
 
==Quantile function==
The formula for the [[quantile function]] is
:<math>
G(p) = \frac{1}{4}\left[\gamma\Phi^{-1}(p) + \sqrt{4+\left(\gamma\Phi^{-1}(p)\right)^2}\right]^2
</math>
where Φ<sup>&nbsp;&minus;1</sup> is the quantile function of the standard normal distribution.
 
==External links==
*[http://www.itl.nist.gov/div898/handbook/eda/section3/eda366a.htm Fatigue life distribution]
 
==References==
 
*{{Citation|last1=Birnbaum|first1= Z. W.|last2=Saunders|authorlink1=Z. W. Birnbaum|first2=S. C.| year=1969| title=A new family of life distributions| journal=Journal of Applied Probability| volume=6| pages=319–327|issue=2|doi=10.2307/3212003| jstor=3212003}}
*{{Citation|last=Desmond| first=A.F.| year=1985| title=Stochastic models of failure in random environments| journal=Canadian Journal of Statistics|volume=13| pages=171–183|issue=3|doi=10.2307/3315148| jstor=3315148}}
*{{Citation|last1=Johnson| first1=N.| last2=Kotz| first2=S.| last3=Balakrishnan| first3=N.| year=1995| title= Continuous Univariate Distributions| volume=2 |edition=2nd| place=New York|publisher=Wiley}}
*{{Citation|journal=Computational Statistics and Data Analysis| volume=51| year=2007| pages=4656–4681| title=Improved statistical inference for the two-parameter Birnbaum&ndash;Saunders distribution |first1=A. J.| last1=Lemonte| first2=F. |last2=Cribari-Neto |first3=K. L. P. |last3=Vasconcellos |doi= 10.1016/j.csda.2006.08.016}}
*{{Citation|journal=Journal of Statistical Computation and Simulation| volume=78| year=2008| pages=37–49| title=Bootstrap-based improved estimators for the two-parameter Birnbaum&ndash;Saunders distribution |first1=A. J.| last1=Lemonte| first2=A. B. |last2=Simas |first3=F. |last3=Cribari-Neto |doi=10.1080/10629360600903882}}
*{{Citation|journal=Computational Statistics and Data Analysis| volume=55| year=2011| pages=1445–1461| title=The β-Birnbaum–Saunders distribution: An improved distribution for fatigue life modeling |first1=G. M.| last1=Cordeiro| first2=A. J. |last2=Lemonte |doi=  10.1016/j.csda.2010.10.007}}
*{{Citation|journal=Brazilian Journal of Probability and Statistics| volume=27| year=2013| pages=133-149| title=A new extension of the Birnbaum–Saunders distribution |first1=A. J.| last1=Lemonte |doi=10.1214/11-BJPS160}}
{{NIST-PD}}
 
{{DEFAULTSORT:Birnbaum-Saunders distribution}}
{{ProbDistributions|continuous-semi-infinite}}
[[Category:Continuous distributions]]
[[Category:Probability distributions]]

Latest revision as of 03:59, 11 January 2015

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