Extreme Value Distribution

For Weibull, let \(Y=\ln(X)\). This has both scale and location parameters. Location is \(\theta_{1}=\ln(\beta)\) and scale is \(\alpha\). Its cdf is:

\begin{equation*} F(x;\theta_{1},\theta_{2})=1-\exp\left(-e^{(x-\theta_{1})/\theta_{2}}\right) \end{equation*}

It is called the extreme value distribution.

Extreme value distributions are the limiting distributions for the minimum or the maximum of a very large collection of random observations from the same arbitrary distribution.

For example, if a system consists of \(n\) identical components in series, and the system fails when the first of these components fails, then system failure times are the minimum of \(n\) random component failure times. Extreme value theory says that, independent of the choice of component model, the system model will approach a Weibull as \(n\) becomes large.

The distribution often referred to as the Extreme Value Distribution (Type I).

(Note that in some places, it lists the cdf as \(1-\) the above. Those are when the maximum instead of the minimum is used).