I got an interesting question about an application in information theory of a statistical
distribution called the Type I extreme value distribution. This distribution, also known as
the Gumbel distribution, is useful for modeling the maximum or minimum of a large number of
variables.
Reliability statistics often consider the maximum or minimum value as an important
quantity to measure. For example, if you have a machine with a large number of components,
and the machine fails when any one of the components fails, then the lifetime of the machine
is the minimum of the individual component failure times. Insurance companies are also
interested in extreme values as they represent the worst case scenarios. A 100 year flood
level also represents an extreme value distribution.
There are two other distributions used to model extreme values (not surprisingly they are
called the Type II and Type III extreme value distributions). Which distribution you use
depends on factors like whether the individual component distribution has finite moments
and/or a bounded tail. I'm not an expert on extreme value problems or reliability statistics,
so I looked up a few good resources.
Extreme Value
Distributions. Annis C. Accessed on 2006-01-09.
[Excerpt] The average of n samples taken from any distribution with finite mean
and variance will have a normal distribution for large n. This is the CLT. The largest
member of a sample of size n has a LEV, Type I largest extreme value, also called Gumbel,
distribution, regardless of the parent population, IF the parent has an unbounded tail
that decreases at least as fast as an exponential function, and has finite moments (as
does the normal, for example). www.statisticalengineering.com/extreme_value.htm
Extreme
value distributions. Tobias P, NIST/SEMATECH e-Handbook of Statistical Methods.
Accessed on 2006-01-09.
[Excerpt] We have already referred to Extreme Value Distributions when describing
the uses of the Weibull 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. Gumbel (1958) showed that for any
well-behaved initial distribution (i.e., F(x) is continuous and has an inverse), only a
few models are needed, depending on whether you are interested in the maximum or the
minimum, and also if the observations are bounded above or below. www.itl.nist.gov/div898/handbook/apr/section1/apr163.htm
Extreme
Value Type I Distribution. Filliben JJ, Heckert A, NIST/SEMATECH e-Handbook of
Statistical Methods. Accessed on 2006-01-09.
[Excerpt] The extreme value type I distribution has two forms. One is based on the
smallest extreme and the other is based on the largest extreme. We call these the
minimum and maximum cases, respectively. Formulas and plots for both cases are given.
The extreme value type I distribution is also referred to as the Gumbel distribution.
www.itl.nist.gov/div898/handbook/eda/section3/eda366g.htm
Extreme value theory.
Wikipedia. Accessed on 2006-01-09.
[Excerpt] Extreme value theory is a branch of statistics dealing with the extreme
deviations from the median of probability distributions. The general theory sets out to
assess the type of probability distributions generated by processes. Extreme value
theory is important for assessing risk for highly unusual events, such as 100-year
floods. en.wikipedia.org/wiki/Extreme_value_theory
A rather unusual approach to extreme values has been advocated by Benoit Mandelbrot,
probably the most famous name in the area of fractal geometry.
Q&A with
Benoit Mandelbrot. Wright CM, National Association of Real Estate Investment
Trusts. Accessed on 2006-01-09.
[Excerpt] Forget Euclidean geometry with its smooth lines and planes. Now comes
Benoit Mandelbrot, the inventor of fractal geometry, who recently wrote an entertaining
and challenging book, "The (mis)Behavior of Markets," in which he argues that his study
of roughness, already applied to topography, meteorology, the compression of computer
files, and many other fields, will rewrite the canon on finance. Portfolio asked the
Yale University mathematics professor, among other things, how real estate prices look
under the fractal microscope. www.nareit.com/portfoliomag/05mayjun/capital.shtml
Mandelbrot's Extremism [PDF]. Beirlant J, Schoutens W, Segers J, Published
December 6, 2004. Accessed on 2006-01-09.
[Abstract] In the sixties Mandelbrot already showed that extreme price swings are
more likely than some of us think or incorporate in our models. A modern toolbox for
analyzing such rare events can be found in the field of extreme value theory. At the
core of extreme value theory lies the modelling of maxima over large blocks of
observations and of excesses over high thresholds. The general validity of these models
makes them suitable for out-of-sample extrapolation. By way of illustration we assess
the likeliness of the crash of the Dow Jones on October 19, 1987, a loss that was more
than twice as large as on any other single day from 1954 until 2004.
www.kuleuven.ac.be/ucs/research/reports/2004/report2004_08.pdf
The work by Mandelbrot may not be useful in the context of the original question, but it
is still a fascinating and very active area of research.
