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Extreme Value Modeling and Risk Analysis: Methods and Applications
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Extreme value theory
Find it at other libraries via WorldCat Limited preview. The first method relies on deriving block maxima minima series as a preliminary step. In many situations it is customary and convenient to extract the annual maxima minima , generating an "Annual Maxima Series" AMS.
The second method relies on extracting, from a continuous record, the peak values reached for any period during which values exceed a certain threshold falls below a certain threshold. For AMS data, the analysis may partly rely on the results of the Fisher—Tippett—Gnedenko theorem , leading to the generalized extreme value distribution being selected for fitting. The theorem here relates to the limiting distributions for the minimum or the maximum of a very large collection of independent random variables from the same distribution.
Given that the number of relevant random events within a year may be rather limited, it is unsurprising that analyses of observed AMS data often lead to distributions other than the generalized extreme value distribution GEVD being selected. For POT data, the analysis may involve fitting two distributions: one for the number of events in a time period considered and a second for the size of the exceedances.
A common assumption for the first is the Poisson distribution , with the generalized Pareto distribution being used for the exceedances.
A tail-fitting can be based on the Pickands—Balkema—de Haan theorem. The field of extreme value theory was pioneered by Leonard Tippett — Tippett was employed by the British Cotton Industry Research Association , where he worked to make cotton thread stronger.
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In his studies, he realized that the strength of a thread was controlled by the strength of its weakest fibres. With the help of R.
Extreme value modeling and risk analysis : methods and applications in SearchWorks catalog
Fisher , Tippet obtained three asymptotic limits describing the distributions of extremes assuming independent variables. Emil Julius Gumbel codified this theory in his book Statistics of Extremes , including the Gumbel distributions that bear his name. These results can be extended to allowing for slight correlations between variables, but the classical theory does not extend to strong correlations of the order of the variance. One universality class of particular interest is that of log-correlated fields , where the correlations decay logarithmically with the distance. A summary of historically important publications relating to extreme value theory can be found in the article List of publications in statistics.
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When normalized, G belongs to one of the following non- degenerate distribution families:. Also known as Type 3.
Also known as Type 1. Also known as Type 2.
Extreme value theory in more than one variable introduces additional issues that have to be addressed.
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