# Extreme Value Distribution

Posted by Beetle B. on Tue 06 June 2017

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).