H22 - H22 Statistics of Extremes There is a long tradition...

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H22 Statistics of Extremes There is a long tradition of phenomenological studies of extreme events in human history. Example 1 Water levels of the Nile. Example 2 Earthquakes. From a statistical perspective , extreme events occur in the tails of probability distributions. In a Gaussian distribution, these tails are exponentials. For many extreme events, the tails are "heavy": for instance, power laws with some fixed power, () , 0 fx x α > . Power laws fall off much slower than exponential (Gaussian) distributions, indicating an enhanced probability of occurrence of extreme events. Let be a sequence of iid random variables with distribution function 12 , , ... XX F and 1 max{ , ..., } nn M = . Of course, , but this result is of no immediate interest, since it simply says that for any fixed ( n n PM x F x ≤= ) x for which , we have () 1 Fx < n x 0 . For a non-trivial limit we must renormalize : find constants and , such that 0 n a > n b n n Mb Px F a x b a ⎛⎞ + ⎜⎟ ⎝⎠ ( ) G x .
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This note was uploaded on 09/23/2009 for the course MATH 200 taught by Professor Gur during the Spring '09 term at Accreditation Commission for Acupuncture and Oriental Medicine.

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H22 - H22 Statistics of Extremes There is a long tradition...

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