Birnbaum-Saunders Distribution

Definition

The Birnbaum-Saunders distribution has the density function

$\frac{1}{\sqrt{2 π}} \exp {- \frac{{(\sqrt{\frac{x}{β}} - \sqrt{\frac{β}{x}})}^{2}}{2 γ^{2}}} (\frac{(\sqrt{\frac{x}{β}} + \sqrt{\frac{β}{x}})}{2 γ x})$

with scale parameter β > 0 and shape parameter γ > 0, for x > 0.

If x has a Birnbaum-Saunders distribution with parameters β and γ, then

$\frac{(\sqrt{\frac{x}{β}} - \sqrt{\frac{β}{x}})}{γ}$

has a standard normal distribution.

Background

The Birnbaum-Saunders distribution was originally proposed as a lifetime model for materials subject to cyclic patterns of stress and strain, where the ultimate failure of the material comes from the growth of a prominent flaw. In materials science, Miner's Rule suggests that the damage occurring after n cycles, at a stress level with an expected lifetime of N cycles, is proportional to n / N. Whenever Miner's Rule applies, the Birnbaum-Saunders model is a reasonable choice for a lifetime distribution model.

Parameters

To estimate distribution parameters, us mle or the Distribution Fitter app.