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The Birnbaum-Saunders distribution represents the distribution of lifetimes for components under certain wear conditions. This unimodal distribution was originally formulated to model failures due to cracks, it is an example of a fatigue life distribution.

The Probability Density Function (PDF) for the Birnbaum-Saunders distribution is [1]

Where α is the shape parameter and β is the scale parameter. The distribution is a mixture of an inverse Gaussian distribution and a reciprocal inverse Gaussian distribution [2]. Shapes of the distribution vary from highly skewed with long tails to almost symmetric as values for β increase [2].

There are several different variations on the formula in the literature. For example, Desmond [3] rewrites the formula as [4]

This formula relaxes some of the assumptions of Birnbaum and Saunders formula, strengthening the physical justification for using the distribution [5]. Rieck and Nedelman [6] presented a log-linear model for the distribution, applicable to accelerated life-testing. If a data set that is thought to be Birnbaum–Saunders distributed, the parameters’ values are best estimated by maximum likelihood [5], although a Jackknife estimator is a possibility [7].

## Properties of the Birnbaum-Saunders Distribution

## References

[1] Birnbaum, Z. W., and Saunders, S. C. (1969). A new family of life distributions, Journal of Applied Probability, 6, 319-327.

[2] Engineering Statistics Handbook. Fatigue Life (Birnbaum-Saunders). Online: https://www.itl.nist.gov/div898/handbook/apr/section1/apr166.htm

[3] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

[4] Desmond, A. F. (1986). On relationship between two fatigue-life models, IEEE Transactions on Reliability, 35, 167-1 69.

[5] Rieck, J. R., and Nedelman, J. (1991). A log-linear model for the Birnbaum-Saunders distribution, Technometrics, 33, 51-60.

[6] Birnbaum, Z. W., and Saunders, S. C. (1969). Estimation for a family of life distributions with applications to fatigue, Journal of Applied Probability, 6, 328-347.

[7] Ahmad, I. A. (1988). Jackknife estimation for a family of life distributions, Journal of Statistical Computation and Simulation, 29, 211-223.