Gumbel and extreme value distributions

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Gumbel and extreme value distributions

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Extreme value distributions are limiting or asymptotic distributions that describe the distribution of the maximum or minimum value drawn from a sample of size n as n becomes large, from an underlying family of distributions (typically the family of Exponential distributions, which includes the Exponential, Gamma, Normal, Weibull and Lognormal). When considering the distribution of minimum values for which a lower bound is known (e.g. there is a lower bound of zero) then the Weibull distribution should be used in preference to the Gumbel. The two distributions are closely related: if X has a Weibull distribution with parameters α and c, then log(X) has an extreme value distribution with parameters µ=log α and β=1/c.

The Gumbel is sometimes referred to as a Log-Weibull, Gompertz or Fisher-Tippett distribution and is a particular case (Type I) of the generalized extreme value distribution. The pdf of the Gumbel distribution is:

with cdf:

The parameter μ is the mode of the distribution, so the sample mode (if easily determined) in conjunction with the sample mean may be used to assist in estimation. Alternatively the median (more easily determined) or maximum likelihood estimation (MLE) methods can be employed. The skewness and kurtosis of the distribution are constants. Sample graphs of the Gumbel pdf are shown below, with key measures provided in the table that follows. We then look at a simple example of the distribution in action.

Gumbel distribution

gumbel

Mean, μ1

μ+βγ

γ is Euler's constant: 0.5772156649...

Mode

μ

 

Median

μ-βk

k=ln(ln(2))

Variance, μ2

(βπ)2/6

 

Skewness

ζ(3) is Apery's constant (from the Riemann zeta function)

Kurtosis

12/5

 

MGF

 

Example: extreme values

In the following example we have taken batches of random samples from a unit Normal. Each batch consisted of 500 random values, and the largest value was then recorded. The process was repeated 1000 times, so a large sample of random extreme (maximum) values was obtained. The frequency distribution of these values was then plotted, as shown below. We know from the Normal distribution that very few values will be 3 or more standard deviations from the mean, but in a sample of 500 there is a reasonable chance that the largest value seen will be greater than 2, and quite a few times values of greater than 3 or 4 will be observed. The histogram reflects this, with a slight positive skew and mode around 3. Also shown is a fitted Gumbel distribution (fitted using iterative methods) confirming its value as a model in such cases. If minimum values were analyzed instead, the same pattern would be observed but with a mode at around -3 and a slight negative skew.

Maximum values from 1000 random unit Normal samples of 500

extreme_Norm_Sample

We can now use this result to model a practical problem. Suppose that a particular chemical compound, A, is used in combination with other substances to produce oral drug tablets. The number of milligrams of compound A is specified and must not exceed a stated maximum value. The manufacturing process produces the drug in batches. A series of initial test batches are produced and the exact chemical makeup is analyzed, with the maximum values in milligrams for compound A in each test batch being recorded. For example, a total of 10 test batches might have been produced. These 10 values are then used to obtain estimates of the parameters of a Gumbel distribution (typically using maximum likelihood estimation) and the probability of obtaining a production batch with greater than the maximum value acceptable can then be calculated from the fitted cumulative distribution. Similar ideas can be applied to estimation of the probability of extreme events, such as damaging waves and storms, floods, and device and materials failures.

To confirm that this kind of result does not just apply to samples from a Normal distribution, we have repeated the exercise but with data from an Exponential distribution with mean=1, so the variance also equals 1. Because this distribution is very long tailed, and bounded by zero from below, only maximum values are considered here, and again the Gumbel distribution provides a good fit as shown below:

Maximum values from 1000 random Exponential samples of 500 with mean value 1

extreme_Exp_Sample

References

[JOH1] Johnson N L, Kotz S (1970) Continuous Univariate Distributions, I. Houghton Mifflin/J Wiley & Sons, New York

[NIST} NIST: eHandbook: Extreme value distributions: http://www.itl.nist.gov/div898/handbook/apr/section1/apr163.htm

Mathworld: Weisstein E W: Gumbel distribution: http://mathworld.wolfram.com/GumbelDistribution.html

Wikipedia: Gumbel distribution: http://en.wikipedia.org/wiki/Gumbel_distribution