Editing Extreme value theory (section)

==Data analysis==
Two main approaches exist for practical extreme value analysis.

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). This method is generally referred to as the ''peak over threshold'' method (POT).<ref>{{cite journal | last = Leadbetter | first = M.R. | year = 1991 | title = On a basis for 'peaks over threshold' modeling | journal = Statistics and Probability Letters | volume = 12 | issue = 4| pages = 357–362 | doi = 10.1016/0167-7152(91)90107-3 }}</ref>

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.<ref>{{harvp|Fisher|Tippett|1928}}</ref><ref>{{harvp|Gnedenko|1943}}</ref> However, in practice, various procedures are applied to select between a wider range of distributions. The theorem here relates to the limiting distributions for the minimum or the maximum of a very large collection of [[statistical independence|independent]] [[random variable]]s 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.<ref>{{harvp|Embrechts|Klüppelberg|Mikosch|1997}}</ref>

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 [[Power law#Estimating the exponent from empirical data|tail-fitting]] can be based on the [[Pickands–Balkema–de Haan theorem]].<ref>{{harvp|Pickands|1975}}</ref><ref>{{harvp|Balkema|de Haan|1974}}</ref>

Novak (2011) reserves the term "POT method" to the case where the threshold is non-random, and distinguishes it from the case where one deals with exceedances of a random threshold.<ref>{{harvp|Novak|2011}}</ref>