Customer Reviews

Extreme Value Theory in Engineering (Statistical Modeling and Decision Science)

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Extreme value theory is very important in the analysis of floods, air pollution, reliability and many other data sources where the maximum or minimum of a sequence of observations is considered. The original development for independent observations goes back to R. A. Fisher and L. Tippetts in the 1920s. After their work Gnedenko (1930s) proved the three types theorem which completed the asymptotic theory for independent and identically distributed random variables. Gumbel wrote the first text on the subject which made it accessible to practitioners. Extensions of the asymptotic theory to correlated sequences were later developed by Berman for Gaussian sequences and Loynes, O'Brien, Leadbetter and others in the more general context of stationary sequences satisfying mixing conditions. My Ph.D. dissertation covered an aspect of the stationary case. Good theoretic treatment can be found in a book by Resnick, one by Galambos and one by Leadbetter, Lindgren and Rootzen. Castillo has collaborated with Galambos in research on extremes. Here he makes the theory and applications accessible to engineers. Even some of my contributions are referenced.

This is one of the best books that I have every read in engineering. Castillo and a few others like Galambos were some of the first people to realize the importance of extreme values. A maximum is an example of an extreme value, because it is the top or highest value. Likewise for a minimum. These extremes behave quite differently from most ordinary things and have a mathematics and engineering all of their own. They have been applied also to other fields where "extremes" in an intuitive sense exist, such as extreme accidents in car insurance (long chain collisions of cars in the fog, heights of waves in the ocean, etc.). They are also closely related to one of the most important new research areas in mathematics: large deviations or rare events theory (see my reviews of Dembo and others). The Gumbel, Weibull, and Frechet probability distributions are important in this area. Those readers who are not specialists should buy the book anyway and hire a consultant or tutor to translate the mathematics into closer to ordinary English, because extreme as well as rare events occur almost everywhere.