The Gumbel distribution is also referred to as the Smallest Extreme Value (SEV) distribution or the Smallest Extreme Value (Type I) distribution. The Gumbel distribution's pdf is skewed to the left, unlike the Weibull distribution's pdf, which is skewed to the right. The Gumbel distribution is appropriate for modeling strength, which is sometimes skewed to the left (few weak units in the lower tail, most units in the upper tail of the strength population). The Gumbel distribution could also be appropriate for modeling the life of products that experience very quick wear-out after reaching a certain age. The distribution of logarithms of times can often be modeled with the Gumbel distribution (in addition to the more common lognormal distribution). [27]
This section includes the following subsections:
The pdf of the Gumbel distribution is given by:
where:
(27)
and:
The Gumbel mean or MTTF is:
where 
(Euler's constant).
The mode of the Gumbel distribution is:
The median of the Gumbel distribution is:
The standard deviation for the Gumbel distribution is given by:
The reliability for a mission of time T for the Gumbel distribution is given by:
The unreliability function is given by:
(28)
The Gumbel reliable life is given by:
The instantaneous Gumbel failure rate is given by:
Some of the specific characteristics of the Gumbel distribution are the following:
The shape of the Gumbel distribution is skewed to the left. The Gumbel pdf has no shape parameter. This means that the Gumbel pdf has only one shape, which does not change.
The Gumbel pdf
has location parameter μ,
which is equal to the mode 
, but it differs from median and
mean. This is because the Gumbel distribution is not symmetrical about
its μ.
As μ decreases, the pdf is shifted to the left.
As μ increases, the pdf is shifted to the right.
As σ increases, the pdf spreads out and becomes shallower.
As σ decreases, the pdf becomes taller and narrower.
For 
pdf = 0. For T
= μ, the pdf
reaches its maximum point (
The points of inflection
of the pdf graph are 
or 
.
If times follow the
Weibull distribution, then the logarithm of times follow a Gumbel distribution.
If ti follows a Weibull
distribution with β
and η,
then the Ln(ti)
follows a Gumbel distribution with μ = ln(η)
and 
[32].
The form of the Gumbel probability paper is based on a linearization of the cdf. From Eqn. (28):
(29)
using Eqn. (27):
Then:
(30)
Now let:
(31)
and:
(32)
(33)
which results in the linear equation of:
The Gumbel probability paper resulting from this linearized cdf function is shown next.
For z
= 0,
T = μ
and 
(63.21% unreliability). For z = 1,
σ
= T - μ
and 
To read μ
from the plot, find the time value that corresponds to the intersection
of the probability plot with the 63.21% unreliability line. To read σ
from the plot, find the time value that corresponds to the intersection
of the probability plot with the 93.40% unreliability line, then take
the difference between this time value and the μ
value.
This section presents the method used by the application to estimate the different types of confidence bounds for data that follow the Gumbel distribution. The complete derivations were presented in detail (for a general function) in the Confidence Bounds chapter. Only Fisher Matrix confidence bounds are available for the Gumbel distribution.
The lower and upper bounds on the mean, 
, are estimated
from:
Since the standard deviation, 
, must be positive, then
ln() is treated as normally
distributed, and the bounds are estimated from:
where Kα is defined by:
If δ
is the confidence level, then 
for the two-sided bounds,
and α = 1 - δ
for the one-sided bounds.
The variances and covariances of and are
estimated from the Fisher matrix as follows:
Λ is the log-likelihood function of the Gumbel distribution, described in the Statistical Background chapter and Appendix C.
The reliability of the Gumbel distribution is given by:
where:
The bounds on z are estimated from:
where:
or:
The upper and lower bounds on reliability are:
The bounds around time for a given Gumbel percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:
where:
or:
The upper and lower bounds are then found by:
Verify using Monte Carlo simulation that if ti follows a Weibull distribution with β and η, then the Ln(ti) follows a Gumbel distribution with μ = ln(η) and σ = 1/β.
Let us assume that ti follows a Weibull distribution with β = 0.5 and η = 10000. The Monte Carlo simulation tool in Weibull++ can be used to generate a set of random numbers that follow a Weibull distribution with the specified parameters.
After obtaining the random time values ti, insert a new Standard Folio. In this Folio enter the Ln(ti) values using the LN function and referring to the cells in the first Folio. Delete any negative values, if there are any, since Weibull++ expects time values to be positive. Calculate the parameters of the Gumbel distribution that fits the Ln(ti) values.
Using maximum likelihood as the analysis method, the estimated parameters are:
Since ln(η) = 9.2103
(
9.3816) and 1/β = 2( 1.9717),
then this simulation verifies that Ln(ti) follows a Gumbel
distribution with μ = ln(η)
and δ
= 1/β.
Note: This example illustrates a property of the Gumbel distribution; it is not meant to be a formal proof.
See Also:
Other Distributions
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