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.

Theory

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

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

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:

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 location parameter.

Properties

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

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

\mu =\beta (1+{\frac {\alpha ^{2}}{2}})

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

\gamma ={\frac {16\alpha ^{2}(11\alpha ^{2}+6)}{(5\alpha ^{2}+4)^{3}}}

\kappa =3+{\frac {6\alpha ^{2}(93\alpha ^{2}+41)}{(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.

Differential equation

The cdf of the Birnbaum-Saunders distribution is a solution of the following differential equation:

\left\{{\begin{array}{l}2\alpha ^{2}\beta x^{2}(\beta +x)f'(x)+f(x)\left(-\beta ^{3}+x^{3}+\left(\alpha ^{2}+1\right)\beta x^{2}+\left(3\alpha ^{2}-1\right)\beta ^{2}x\right)=0,\\[12pt]f(1)={\frac {(\beta +1)e^{{-{\frac {(\beta -1)^{2}}{2\alpha ^{2}\beta }}}}}{2{\sqrt {2\pi }}\alpha {\sqrt {\beta }}}}\end{array}}\right\}

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.

Transformation

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

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

Equivalently

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.

Probability density function

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

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 $\phi$ 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

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.

Cumulative distribution function

The formula for the cumulative distribution function is

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.

Quantile function

The formula for the quantile function is

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.

External links

Fatigue life distribution

References

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
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, 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: 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: 1445–1461, doi:10.1016/j.csda.2010.10.007
Lemonte, A. J. (2013), "A new extension of the Birnbaum–Saunders distribution", Brazilian Journal of Probability and Statistics, 27: 133–149, doi:10.1214/11-BJPS160

This article incorporates public domain material from the National Institute of Standards and Technology website http://www.nist.gov.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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