Birnbaum-Saunders Distribution

R2026b

Definition

The Birnbaum-Saunders distribution has the density function

$$\frac{1}{\sqrt{2\pi }}\exp \left\{-\frac{{\left(\sqrt{\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.}-\sqrt{\raisebox{1ex}{$\beta $}\!\left/ \!\raisebox{-1ex}{$x$}\right.}\right)}^2}{2{\gamma }^2}\right\}\left(\frac{\left(\sqrt{\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.}+\sqrt{\raisebox{1ex}{$\beta $}\!\left/ \!\raisebox{-1ex}{$x$}\right.}\right)}{2\gamma x}\right)$$

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

$$\frac{\left(\sqrt{\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.}-\sqrt{\raisebox{1ex}{$\beta $}\!\left/ \!\raisebox{-1ex}{$x$}\right.}\right)}{\gamma }$$

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

The Birnbaum-Saunders distribution uses the following parameters.

Parameter	Description	Support
β	Scale parameter	β > 0
γ	Shape parameter	γ > 0

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