Birnbaum-Saunders distribution

The Birnbaum-Saunders distribution (also known as the fatigue life distribution) is used extensively in reliability applications to model failure times. There are several alternative formulations of this distribution in the literature.

Probability density function

The general formula for the probability density function (pdf) is:where γ is the shape parameter, μ is the location parameter, β is the scale parameter, and $phi\; ,$ is the probability density function of the standard normal distribution.

tandard 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:$f(x)\; =\; frac\{sqrt\{x\}+sqrt\{frac\{1\}\{x\}\{2gamma\; x\}phileft(frac\{sqrt\{x\}-sqrt\{frac\{1\}\{x\}\{gamma\}\; ight)quad\; x\; >\; 0;\; gamma\; >0$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.

Cumulative distribution function

The formula for the cumulative distribution function is:$F(x)\; =\; Phileft(frac\{sqrt\{x\}\; -\; sqrt\{frac\{1\}\{x\}\{gamma\}\; ight)quad\; x\; >\; 0;\; gamma\; >\; 0$

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

Quantile function

The formula for the quantile function is:$G(p)\; =\; frac\{1\}\{4\}left\; [gammaPhi^\{-1\}(p)\; +\; sqrt\{4+left(gammaPhi^\{-1\}(p)\; ight)^2\}\; ight]\; ^2$where Φ ⁻¹ 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]

Further reading

*Citation|last1=Birnbaum|first1= Z.W|last2=Saunders|first2=S.C.| year=1969| title=A new family of life distributions| journal=Journal of Applied Probability| volume=6| pages=319-327|url=http://links.jstor.org/sici?sici=0021-9002%28196908%296%3A2%3C319%3AANFOLD%3E2.0.CO%3B2-9
*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|url=http://links.jstor.org/sici?sici=0319-5724%28198509%2913%3A3%3C171%3ASMOFIR%3E2.0.CO%3B2-W
*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-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-Saunders distribution |first1=A. J.| last1=Lemonte| first2=A. B. |last2=Simas |first3=F. |last3=Cribari-Neto |doi=10.1080/10629360600903882