Gumbel Distribution for Extreme Values ◾ 329
Review Questions
1. When should we use the Gumbel minimum distribution?
2. When should we use the Gumbel maximum distribution?
3. What is the primary use of extreme value theory?
4. Relate outliers in a box plot to Gumbel distributions.
5. Compare the bell curve with the Gumbel distribution.
Box 20.5 LIVING oN thE EDGE
Combine the extremes, and you will have the true center.
Karl Wilhelm Friedrich Schlegel
Which one captures the essential feature of life: central tendency, dispersion,
or extreme values? Extreme values have crossed the boundary; they contain
novelty and throb with energy. ey have been pushed by extreme circum-
stances and cruel experiments by nature. We can learn infinitely more and
gain infinitely more from extreme data than from regular data.
Life is in the boundary.
We have lived out of central limit theorem, and now we should reach out
and consider extreme value theorem. Both theorems have been designed to
tell us about limiting performances. Let us look at some of the problems
modeled by extreme value theory since Tippett and Gumbel: extreme for-
est fire, extremely high flood levels, extreme heights of waves, earthquake,
extreme heat, extreme cold, extreme loads on aircraft structure, excessive
stock movements, extreme load on wind turbines, and the list is growing. All
these extreme value studies aim to save lives or property.
To solve a problem, look at the problem.
However, the optimism of Extreme Value eory (EVT) is appropriate
but also exaggerated sometimes. Yet it holds great promise. e potential of
EVT remains latent, much so in software engineering practices.
330 ◾ Simple Statistical Methods for Software Engineering
Exercises
1. Program Gumbel minimum distribution in Excel. (ere is no readily avail-
able function in Excel.)
2. Program Gumbel maximum distribution in Excel. (ere is no readily avail-
able function in Excel.)
3. In the CSAT Gumbel model with location = 3 and scale = 1, find the risk of
CSAT score falling below 2. Make use the program you have developed for
exercise 1.
4. In the complexity Gumbel maximum model with location = 169 and scale =
87, find the risk of complexity exceeding 300. Make use of the program you
have developed for exercise 1.
5. For schedule variance Gumbel maximum model with location parameter =
20 and scale parameter = 12, find the risk of schedule variance exceeding 40.
References
1. C. Neves and M. I. Fraga Alves, Testing extreme value conditions—An overview and
recent approaches, REVSTAT: Statistical Journal, 6(1), 83–100, 2008.
2. Engineering Statistics Handbook, NIST. http://www.itl.nist.gov/div898/handbook/.
3. J. H. J. Einmahl and S. G. W. R. Smeets, Ultimate 100m World Records rough Extreme-
Value eory, Tilburg University, AZL, Heerlen, July 10, 2009, ISSN 0924-7815.
4. A. A. Zimbidis et al., Modeling earthquake risk via extreme value theory and pric-
ing the respective catastrophe bonds, Astin Bulletin, 37(1), 163–183. doi: 10.2143/
AST.37.1.2020804
©
2007.
5. Y. Lu, T. Nolt, I. Bate and L. Cucu-Grosjean, A Trace-Based Statistical Worst-Case
Execution Time Analysis of Component-Based Real-Time Embedded Systems. IEEE,
Emerging Technologies & Factory Automation (ETFA), 2011 IEEE 16th Conference,
pp. 1–4, Toulouse, 2011.
6. C. Maxim, A. Gogonel, D. Maxim and L. Cucu-Grosjean, Estimation of Probabilistic
Minimum Inter-Arrival Times Using Extreme Value eory, INRIA Nancy-Grand Est,
France. https://hal.inria.fr/hal-00766063/PDF/JW_2012_submitted_26september_v2
.pdf.
7. S. Caires, Extreme Value Analysis: Wave Data, 2011. http://www.jodc.go.jp/info/ioc_doc
/JCOMM_Tech/JCOMM-TR-057.pdf.
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