1.3. EDA Techniques
1.3.6. Probability Distributions
1.3.6.6. Gallery of Distributions
1.3.6.6.10. |
Birnbaum-Saunders (Fatigue Life) Distribution |
The general formula for the probability density function of the Birnbaum-Saunders distribution is
\( f(x) = \left (\frac{\sqrt{\frac{x-\mu} {\beta}} + \sqrt{\frac{\beta} {x-\mu}}} {2\gamma (x-\mu)} \right) \phi \left( \frac{\sqrt{\frac{x-\mu} {\beta}} - \sqrt{\frac{\beta} {x-\mu}}} {\gamma} \right) \hspace{.2in} x > \mu; \gamma, \beta > 0 \)
where γ is the shape parameter, μ is the location parameter, β is the scale parameter, \(\phi\) is the probability density function of the standard normal distribution, and \(\Phi\) is the cumulative distribution function of the standard normal distribution. The case where μ = 0 and β = 1 is called the standard Birnbaum-Saunders distribution. The equation for the standard Birnbaum-Saunders distribution reduces to
\( f(x) = \left (\frac{\sqrt{x} + \sqrt{\frac{1} {x}}} {2\gamma x} \right) \phi \left (\frac{\sqrt{x} - \sqrt{\frac{1} {x}}} {\gamma} \right) \hspace{.2in} x > 0; \gamma > 0 \)
Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.
The following is the plot of the Birnbaum-Saunders probability density function.
\( F(x) = \Phi(\frac{\sqrt{x} - \sqrt{\frac{1} {x}}} {\gamma}) \hspace{.2in} x > 0; \gamma > 0 \)
where \(\Phi\) is the cumulative distribution function of the standard normal distribution. The following is the plot of the Birnbaum-Saunders cumulative distribution function with the same values of γ as the pdf plots above.
\( G(p) = \frac{1} {4} \left[\gamma \Phi^{-1}(p) + \sqrt{4 + (\gamma \Phi^{-1}(p))^{2}}\right]^{2} \)
where \(\Phi^{-1}\) is the percent point function of the standard normal distribution. The following is the plot of the Birnbaum-Saunders percent point function with the same values of γ as the pdf plots above.
The following is the plot of the Birnbaum-Saunders hazard function with the same values of γ as the pdf plots above.
The following is the plot of the Birnbaum-Saunders cumulative hazard function with the same values of γ as the pdf plots above.
The following is the plot of the Birnbaum-Saunders survival function with the same values of γ as the pdf plots above.
The following is the plot of the gamma inverse survival function with the same values of γ as the pdf plots above.
| Mean | \( 1 + \frac{\gamma^{2}} {2} \) |
| Range | 0 to \(\infty\). |
| Standard Deviation | \( \gamma\sqrt{1 + \frac{5\gamma^{2}} {4}} \) |
| Coefficient of Variation | \( \frac{2 + \gamma^{2}} {\gamma\sqrt{1 + 5\gamma^{2}}} \) |
The "bs" package implements support for the Birnbaum-Saunders distribution for the R package. See
-
Leiva, V., Hernandez, H., and Riquelme, M. (2006). A New Package
for the Birnbaum-Saunders Distribution. Rnews, 6/4, 35-40.
(http://www.r-project.org)
