Extreme Value Distributions
Three types of asymptotic distributions have been
developed for maximum and minimum values based on
different initial distributions. These distributions are
based on the extreme types theorem, and they are widely
used in risk management, finance, economics, material
science and other industries. In this article, we will
introduce all three distributions. Two of them are of
particular interest in
reliability engineering: the type I asymptotic
distribution for both maximum and minimum values (for
minimum values,
this is referred to as the Gumbel/Smallest Extreme
Value (SEV) distribution in
Weibull++),
and the type III asymptotic distribution for minimum
values (i.e., the wellknown Weibull distribution).
Type I Distribution
The extreme value type I distribution has two forms.
One is based on the largest extreme and the other is
based on the smallest extreme. These two forms of the
distribution can be used to model the distribution of
the maximum or minimum number of the samples of various
distributions. For example, if you had a list of maximum
river levels for each of the past ten years, you could
use the extreme value type I distribution to represent
the distribution of the maximum level of a river in an
upcoming year. This distribution is particularly useful
in predicting the chance that an earthquake,
flood or other natural disaster will occur.
The general formula for the pdf of the type I
(minimum) distribution is:
where:
 μ is the location parameter.
 σ is the scale parameter.
When μ=0, σ=1, the above
equation reduces to the standard Gumbel
(minimum) distribution:
The reliability
function of the Gumbel (minimum) distribution is given
by:
The general formula for the pdf of
the type I (maximum) distribution is:
When μ=0, σ=1, the above equation reduces to the
standard Gumbel (maximum) distribution:
The reliability function of the Gumbel
(maximum) distribution is given by:
Type II Distribution
The extreme value type II distribution, also known as
the Fréchet distribution, is defined as:
where:
 α is the shape parameter.
 β is the scale parameter.
The reliability function
of the extreme value type II is given by:
Type III Distribution
The extreme value type III distribution for minimum
values is the wellknown Weibull distribution. The twoparameter
Weibull distribution is given by:
where:
 α is the shape parameter.
 β is the scale parameter.
The reliability function for the Weibull
distribution is given by:
Example
This section provides an example to show how
to use the Gumbel/SEV distribution to solve a problem in
Weibull++.
A sample of 40 bearings from the same
production lot is tested. These 40 bearings are randomly
divided into 5 groups of 8 bearings each. The bearings
in each group are tested simultaneously, and the test of
each group is terminated when one bearing in that group
fails. The times to the first failure of each group
are given in Table 1.
Table 1  Time to first failure for each of 5 groups of 8 bearings
Order of Failure 
Time to First Failure 
1 
65 
2 
120 
3 
155 
4 
200 
5 
300 
The
times to first failure may be
represented by the extreme value type I distribution for
minimum values, so the Gumbel/SEV
distribution can be used for calculation in Weibull++.
Enter the data from Table 1 into a Weibull++
standard folio using a data sheet for times to failure
with no censoring and no grouping. Then calculate the
data sheet using the Gumbel distribution and the RRX
parameter estimation method, as shown in Figure 1:
Figure 1  Gumbel/SEV
distribution in Weibull++
The calculated location
parameter, μ, is 208.3001, and the scale parameter,
σ, is
equal to 82.0319.
Based on this, the reliability of a product with
8 bearings in series for a mission of 50 hours with a 90%
lower onesided confidence bound can be obtained using the QCP, as
shown in Figure 2.
Figure 2
 Reliability calculation in
QCP
Conclusion
In the article, we reviewed three types of extreme
value distributions. The type I asymptotic distribution and
the type III asymptotic distribution for minimum values are
widely used in reliability engineering. The extreme
value type I distribution
has two forms: the smallest extreme (which is implemented in Weibull++ as the Gumbel/SEV
distribution) and the largest extreme. The extreme value
type III distribution for minimum values is
actually the Weibull distribution. For more about the Weibull
distribution, please see
http://reliawiki.org/index.php/The_Weibull_Distribution.
References
Kececioglu, Dmitri, Reliability Engineering
Handbook, Volume 1, Englewood Cliffs, NJ:
PrenticeHall, Inc., 1991.
