Editing Extreme value theory

{{Short description|Branch of statistics focusing on large deviations}}
{{About|the statistical theory|the result in calculus|extreme value theorem}}
[[File:1755 Lisbon earthquake.jpg|thumb|upright=1.78|Extreme value theory is used to model the risk of extreme, rare events, such as the [[1755 Lisbon earthquake]].]]

'''Extreme value theory''' or '''extreme value analysis''' ('''EVA''') is the study of extremes in statistical distributions.

It is widely used in many disciplines, such as [[structural engineering]], [[finance]], [[economics]], [[earth science]]s, traffic prediction, and [[Engineering geology|geological engineering]]. For example, EVA might be used in the field of [[hydrology]] to estimate the probability of an unusually large flooding event, such as the [[100-year flood]]. Similarly, for the design of a [[breakwater (structure)|breakwater]], a [[coastal engineer]] would seek to estimate the 50&nbsp;year wave and design the structure accordingly.

==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>

==Applications==
Applications of extreme value theory include predicting the probability distribution of:
{{div col begin |colwidth = 20em}}
* Extreme [[flood]]s; the size of [[freak wave]]s
* [[Tornado]] outbreaks<ref>{{harvp|Tippett|Lepore|Cohen|2016}}</ref>
* Maximum sizes of [[population (ecology)|ecological population]]s<ref>
{{cite journal
 |last1=Batt      |first1=Ryan D.
 |last2=Carpenter |first2=Stephen R.
 |last3=Ives      |first3=Anthony R.
 |date=March 2017
 |title=Extreme events in lake ecosystem time series
 |journal=Limnology and Oceanography Letters
 |volume=2 |issue=3 |page=63
 |doi=10.1002/lol2.10037  |doi-access=free
 |bibcode=2017LimOL...2...63B
}}
</ref>
* [[Side effect]]s of [[prescription drug|drugs]] (e.g., [[ximelagatran]])
* The magnitudes of large [[insurance]] losses
* [[Equity risk]]s; day-to-day [[market risk]]
* [[Mutation]] events during [[evolution]]
* Large [[wildfire]]s<ref>{{harvp|Alvarado|Sandberg|Pickford|1998|p=68}}</ref>
* [[Structural load|Environmental loads on structures]]<ref>{{harvp|Makkonen|2008}}</ref>
* Time the fastest [[human running speed|humans could ever run]] the [[100 metres|100 metres sprint]]<ref>
{{cite report
 |first1=J.H.J.   |last1=Einmahl
 |first2=S.G.W.R. |last2=Smeets
 |year=2009
 |title=Ultimate 100m world records through extreme-value theory
 |series=CentER Discussion Paper |volume=57
 |publisher=Tilburg University
 |url=https://pure.uvt.nl/ws/files/1244969/j.1467-9574.2010.00470.x.pdf
 |access-date=2009-08-12
 |archive-url=https://web.archive.org/web/20160312023048/https://pure.uvt.nl/ws/files/1244969/j.1467-9574.2010.00470.x.pdf
 |archive-date=2016-03-12
}}
</ref> and performances in other [[Sport of athletics|athletic]] disciplines<ref>
{{cite journal
 |first1=D. |last1=Gembris
 |first2=J. |last2=Taylor
 |first3=D. |last3=Suter
 | title = Trends and random fluctuations in athletics
 | journal = Nature
 | volume = 417  | issue =  6888 | page = 506
 | year = 2002
 | doi=10.1038/417506a | doi-access = free
 | bibcode =  2002Natur.417..506G | pmid=12037557
 | hdl =  2003/25362 | s2cid = 13469470
}}
</ref><ref>
{{cite journal
 | first1 = D. | last1 = Gembris
 | first2 = J. | last2 = Taylor
 | first3 = D. | last3 = Suter
 | year = 2007
 | title = Evolution of athletic records: Statistical effects versus real improvements
 | journal = [[Journal of Applied Statistics]]
 | volume = 34 | issue = 5 | pages = 529–545
 | doi=10.1080/02664760701234850  | pmid = 38817921
 | s2cid = 55378036
 | bibcode=2007JApSt..34..529G  | hdl =  2003/25404
| pmc = 11134017}}
</ref><ref>
{{cite journal
 | first1 = H.  | last1 = Spearing        | first2 = J.  | last2 = Tawn
 | first3 = D.  | last3 = Irons           | first4 = T.  | last4 = Paulden
 | first5 = G.  | last5 = Bennett 
 | year = 2021
 | title = Ranking, and other properties, of elite swimmers using extreme value theory
 | journal = Journal of the Royal Statistical Society
 | series = Series&nbsp;A (Statistics in Society)
 | volume = 184  | issue = 1  | pages = 368–395
 | doi=10.1111/rssa.12628  | doi-access = free
 | s2cid = 204823947
| arxiv = 1910.10070}}
</ref>
* Pipeline failures due to [[pitting corrosion]]
* Anomalous IT [[network traffic]], prevent [[security hacker|attackers]] from reaching important data
* [[Road safety]] analysis<ref>
{{cite journal
 |last1=Songchitruksa |first1=P.
 |last2=Tarko         |first2=A.P.
 |year =2006
 |title=The extreme value theory approach to safety estimation
 |journal=Accident Analysis and Prevention
 |volume=38  |issue=4  |pages=811–822
 |doi=10.1016/j.aap.2006.02.003  |pmid=16546103
}}
</ref><ref>
{{cite journal
 |last1=Orsini   |first1=F.       |last2=Gecchele |first2=G.
 |last3=Gastaldi |first3=M.       |last4=Rossi    |first4=R.
 |date=2019
 |title=Collision prediction in roundabouts: A comparative study of extreme value theory approaches
 |journal=Transportmetrica
 |series=Series&nbsp;A: Transport Science
 |volume=15  |issue=2  |pages=556–572
 |doi=10.1080/23249935.2018.1515271
 |s2cid=158343873
}}
</ref>
* [[Wireless communications]]<ref>
{{cite journal
 |first1=C.G. |last1=Tsinos           |first2=F.  |last2=Foukalas
 |first3=T.   |last3=Khattab          |first4=L.  |last4=Lai
 |date=February 2018
 |title=On channel selection for carrier aggregation systems
 |journal=[[IEEE Transactions on Communications]]
 |volume=66  |issue=2  |pages=808–818
 |doi=10.1109/TCOMM.2017.2757478 |s2cid=3405114 |url=https://ieeexplore.ieee.org/document/8052574
}}
</ref>
* [[Epidemic]]s<ref>
{{cite journal
 |last1=Wong    |first1=Felix
 |last2=Collins |first2=James J.
 |date=2020-11-02  |df=dmy-all
 |title=Evidence that coronavirus superspreading is fat-tailed
 |journal=Proceedings of the National Academy of Sciences of the USA
 |volume=117  |issue=47  |pages=29416–29418
 |doi=10.1073/pnas.2018490117  |doi-access=free
 |issn=0027-8424  |pmid=33139561  |pmc=7703634
 |bibcode=2020PNAS..11729416W
}}
</ref>
* [[Neurobiology]]<ref>
{{Cite journal
 |last1=Basnayake  |first1=Kanishka       |last2=Mazaud  |first2=David
 |last3=Bemelmans  |first3=Alexis         |last4=Rouach  |first4=Nathalie
 |last5=Korkotian  |first5=Eduard         |last6=Holcman |first6=David
 |date=2019-06-04  |df=dmy-all
 |title=Fast calcium transients in dendritic spines driven by extreme statistics
 |journal=[[PLOS Biology]]
 |volume=17  |issue=6  |page=e2006202
 |doi=10.1371/journal.pbio.2006202 |doi-access=free 
 |pmid=31163024  |issn=1545-7885  |pmc=6548358
}}</ref>
* [[Solar energy]]<ref>
{{cite journal
 |last1=Younis     |first1=Abubaker
 |last2=Abdeljalil |first2=Anwar
 |last3=Omer       |first3=Ali
 |date=2023-01-01  |df=dmy-all
 |title=Determination of panel generation factor using peaks over threshold method and short-term data for an off-grid photovoltaic system in Sudan: A case of Khartoum city
 |journal=Solar Energy
 |volume=249  |pages=242–249
 |doi=10.1016/j.solener.2022.11.039
 |bibcode=2023SoEn..249..242Y
 |s2cid=254207549  |issn=0038-092X
 |url=https://www.sciencedirect.com/science/article/pii/S0038092X22008593
}}
</ref>
* Extreme [[Space weather]]<ref>{{cite journal |last1=Fogg |first1=Alexandra Ruth |title=Extreme Value Analysis of Ground Magnetometer Observations at Valentia Observatory, Ireland |journal=Space Weather |date=2023 |volume=21 |issue=e2023SW003565 |doi=10.1029/2023SW003565 |bibcode=2023SpWea..2103565F |url=https://doi.org/10.1029/2023SW003565}}</ref><ref>{{cite journal |last1=Elvidge |first1=Sean |title=Estimating the occurrence of geomagnetic activity using the Hilbert-Huang transform and extreme value theory. |journal=Space Weather |date=2020 |volume=17 |issue=e2020SW002513 |doi=10.1029/2020SW002513 |doi-access=free |bibcode=2020SpWea..1802513E }}</ref><ref>{{cite journal |last1=Bergin |first1=Aisling |title=Extreme event statistics in Dst, SYM-H, and SMR geomagnetic indices |journal=Space Weather |date=2023 |volume=21 |issue=e2022SW003304 |doi=10.1029/2022SW003304 |bibcode=2023SpWea..2103304B |url=https://doi.org/10.1029/2022SW003304|hdl=10037/30641 |hdl-access=free }}</ref>
{{div col end}}

==History==
The field of extreme value theory was pioneered by [[Leonard Tippett|L. Tippett]] (1902–1985). Tippett was employed by the [[British Cotton Industry Research Association]], where he worked to make cotton thread stronger. In his studies, he realized that the strength of a thread was controlled by the strength of its weakest fibres. With the help of [[Ronald Fisher|R.A. Fisher]], Tippet obtained three asymptotic limits describing the distributions of extremes assuming independent variables. [[Emil Julius Gumbel|E.J. Gumbel]] (1958)<ref>{{harvp|Gumbel|2004}}</ref> codified this theory. These results can be extended to allow 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.

==Univariate theory==
{{main|Fisher–Tippett–Gnedenko theorem|l1=Extreme value theorem}}
The theory for extreme values of a single variable is governed by the ''[[Fisher–Tippett–Gnedenko theorem|extreme value theorem]]'', also called the ''[[Fisher–Tippett–Gnedenko theorem]]'', which describes which of the three possible distributions for extreme values applies for a particular statistical variable <math>X</math>.

==Multivariate theory==
Extreme value theory in more than one variable introduces additional issues that have to be addressed. One problem that arises is that one must specify what constitutes an extreme event.<ref name=Morton-Bowers-1996>
{{cite journal
 |last1=Morton |first1=I.D.
 |last2=Bowers |first2=J.
 |date=December 1996
 |title=Extreme value analysis in a multivariate offshore environment
 |journal=Applied Ocean Research
 |volume=18 |issue=6 |pages=303–317
 |doi=10.1016/s0141-1187(97)00007-2
 |bibcode=1996AppOR..18..303M
 |issn=0141-1187
}}
</ref>
Although this is straightforward in the univariate case, there is no unambiguous way to do this in the multivariate case. The fundamental problem is that although it is possible to order a set of real-valued numbers, there is no natural way to order a set of vectors.

As an example, in the univariate case, given a set of observations <math>\ x_i\ </math> it is straightforward to find the most extreme event simply by taking the maximum (or minimum) of the observations. However, in the bivariate case, given a set of observations <math>\ ( x_i, y_i )\ </math>, it is not immediately clear how to find the most extreme event. Suppose that one has measured the values <math>\ (3, 4)\ </math> at a specific time and the values <math>\ (5, 2)\ </math> at a later time. Which of these events would be considered more extreme? There is no universal answer to this question.

Another issue in the multivariate case is that the limiting model is not as fully prescribed as in the univariate case. In the univariate case, the model ([[Generalized extreme value distribution|GEV distribution]]) contains three parameters whose values are not predicted by the theory and must be obtained by fitting the distribution to the data. In the multivariate case, the model not only contains unknown parameters, but also a function whose exact form is not prescribed by the theory. However, this function must obey certain constraints.<ref>
{{cite book
 |last1=Beirlant |first1=Jan    |last2=Goegebeur |first2=Yuri
 |last3=Teugels  |first3=Jozef  |last4=Segers    |first4=Johan
 |date=2004-08-27 |df=dmy-all
 |title=Statistics of Extremes: Theory and applications
 |publisher=John Wiley & Sons, Ltd
 |series=Wiley Series in Probability and Statistics
 |location=Chichester, UK
 |doi=10.1002/0470012382 |isbn=978-0-470-01238-3
}}
</ref><ref>
{{cite book
 |last=Coles |first=Stuart
 |year=2001
 |title=An Introduction to Statistical Modeling of Extreme Values
 |series=Springer Series in Statistics
 |doi=10.1007/978-1-4471-3675-0
 |issn=0172-7397 |isbn=978-1-84996-874-4
}}
</ref>
It is not straightforward to devise estimators that obey such constraints though some have been recently constructed.<ref name=dC2014>
{{cite journal
 |last1=de&nbsp;Carvalho |first1=M.
 |last2=Davison |first2=A.C.
 |year=2014
 | title = Spectral density ratio models for multivariate extremes
 |journal=Journal of the American Statistical Association
 |volume=109 |pages=764‒776
 |s2cid=53338058  |doi=10.1016/j.spl.2017.03.030
 |hdl=20.500.11820/9e2f7cff-d052-452a-b6a2-dc8095c44e0c
 |url = https://www.maths.ed.ac.uk/~mdecarv/papers/decarvalho2014a.pdf
}}
</ref><ref name=hanson2017>
{{cite journal
 |last1=Hanson           |first1=T.
 |last2=de&nbsp;Carvalho |first2=M.
 | last3=Chen            |first3=Yuhui
 | title=Bernstein polynomial angular densities of multivariate extreme value distributions
 |journal=Statistics and Probability Letters
 |year=2017
 |volume=128 |pages=60–66
 |doi=10.1016/j.spl.2017.03.030  |s2cid=53338058
 |hdl=20.500.11820/9e2f7cff-d052-452a-b6a2-dc8095c44e0c
 |url = https://www.maths.ed.ac.uk/~mdecarv/papers/hanson2017.pdf
}}</ref><ref name=dC2013>
{{cite journal
 |last1=de&nbsp;Carvalho |first1=M.
 |year=2013
 | title = A Euclidean likelihood estimator for bivariate tail dependence
 | journal=Communications in Statistics – Theory and Methods
 |volume=42 |issue=7 |pages=1176–1192
 | doi= 10.1080/03610926.2012.709905
 |arxiv=1204.3524 |s2cid=42652601
 |url = https://www.maths.ed.ac.uk/~mdecarv/papers/decarvalho2013.pdf
}}
</ref>

As an example of an application, bivariate extreme value theory has been applied to ocean research.<ref name=Morton-Bowers-1996/><ref>
{{cite journal
 |last1=Zachary |first1=S.        |last2=Feld    |first2=G.
 |last3=Ward    |first3=G.        |last4=Wolfram |first4=J.
 |date=October 1998
 |title=Multivariate extrapolation in the offshore environment
 |journal=[[Applied Ocean Research]]
 |volume=20  |issue=5  |pages=273–295
 |doi=10.1016/s0141-1187(98)00027-3
 |bibcode=1998AppOR..20..273Z |issn=0141-1187
}}
</ref>

==Non-stationary extremes==
Statistical modeling for nonstationary time series was developed in the 1990s.<ref name=dS1990>
{{cite journal
 |last1 = Davison |first1 = A.C.
 |last2 = Smith   |first2 = Richard
 |year=1990
 |title = Models for exceedances over high thresholds
 |journal=Journal of the Royal Statistical Society
 |series=Series&nbsp;B (Methodological)
 |volume=52  |issue=3  |pages=393–425
 |doi= 10.1111/j.2517-6161.1990.tb01796.x
 |url = https://rss.onlinelibrary.wiley.com/doi/10.1111/j.2517-6161.1990.tb01796.x
}}
</ref> Methods for nonstationary multivariate extremes have been introduced more recently.<ref name=dC2012>
{{cite book
 |last=de&nbsp;Carvalho |first=M.
 |year=2016
 |section=Statistics of extremes: Challenges and opportunities
 |title=Handbook of EVT and its Applications to Finance and Insurance
 |location=Hoboken, NJ
 |publisher=John Wiley's Sons
 |pages=195–214
 |isbn=978-1-118-65019-6
 |url=https://www.maths.ed.ac.uk/~mdecarv/papers/decarvalho2016b.pdf
}}
</ref>
The latter can be used for tracking how the dependence between extreme values changes over time, or over another covariate.<ref name=castro2018>
{{cite journal
 |first1 = D. |last1 = Castro
 |first2 = M. |last2 = de&nbsp;Carvalho
 |first3 = J. |last3 = Wadsworth
 |year = 2018
 |title = Time-Varying Extreme Value Dependence with Application to Leading European Stock Markets
 |journal=Annals of Applied Statistics
 |volume=12  |pages=283–309
 |doi= 10.1214/17-AOAS1089  |s2cid=33350408
 |url = https://www.maths.ed.ac.uk/~mdecarv/papers/castro2018.pdf
}}
</ref><ref name=mhalla2019>
{{Cite journal
 |last1 = Mhalla            |first1=L.
 |last2 = de&nbsp;Carvalho  |first2 = M.
 |last3 = Chavez-Demoulin   |first3 = V.
 |year=2019
 |title = Regression type models for extremal dependence
 |journal=Scandinavian Journal of Statistics
 |volume=46  |issue=4  |pages=1141–1167
 | doi= 10.1111/sjos.12388  |s2cid=53570822
 |url = https://www.maths.ed.ac.uk/~mdecarv/papers/mhalla2019.pdf
}}
</ref><ref name=EB2018>
{{cite journal
 |last1 = Mhalla           |first1 = L.
 |last2 = de&nbsp;Carvalho |first2 = M.
 |last3 = Chavez-Demoulin  |first3 = V.
 |year=2018
 |title = Local robust estimation of the Pickands dependence function
 |journal=[[Annals of Statistics]]
 |volume=46  |issue=6A  |pages=2806–2843
 |s2cid=59467614
 |doi=10.1214/17-AOS1640 |doi-access=free
}}
</ref>

==See also==
{{div col begin | colwidth = 20em }}
* [[Extreme risk]]
* [[Extreme weather]]
* [[Fisher–Tippett–Gnedenko theorem]]
* [[Generalized extreme value distribution]]
* [[Large deviation theory]]
* [[Outlier]]
* [[Pareto distribution]]
* [[Pickands–Balkema–de Haan theorem]]
* [[Rare events]]
* [[Redundancy principle (biology)|Redundancy principle]]
; Extreme value distributions
* [[Fréchet distribution]]
* [[Gumbel distribution]]
* [[Weibull distribution]]
{{div col end}}

<!--
{{more footnotes needed|date=September 2010}}
-->

==References==
{{reflist|25em}}

==Sources==
{{refbegin|colwidth=25em|small=yes}}
* {{cite journal
 | last1 = Abarbanel  | first1 = H.     | last2 = Koonin    | first2 = S.
 | last3 = Levine     | first3 = H.     | last4 = MacDonald | first4 = G.
 | last5 = Rothaus    | first5 = O.
 | date = January 1992
 | title = Statistics of extreme events with application to climate
 | journal = JASON
 | volume = JSR-90-30S
 | url = http://www.fas.org/irp/agency/dod/jason/statistics.pdf
 | access-date = 2015-03-03
}}
* {{cite journal
 | last1 = Alvarado   | first1 = Ernesto
 | last2 = Sandberg   | first2 = David V.
 | last3 = Pickford   | first3 = Stewart G.
 | year = 1998
 | title = Modeling Large Forest Fires as Extreme Events
 | journal = [[Northwest Science]]
 | volume = 72 | pages = 66–75
 | url = http://www.vetmed.wsu.edu/org_nws/NWSci%20journal%20articles/1998%20files/Special%20addition%201/v72%20p66%20Alvarado%20et%20al.PDF
 | access-date = 2009-02-06
 | archive-url = https://web.archive.org/web/20090226080558/http://www.vetmed.wsu.edu/org_nws/NWSci%20journal%20articles/1998%20files/Special%20addition%201/v72%20p66%20Alvarado%20et%20al.PDF
 | archive-date = 2009-02-26
}}
* {{cite journal
 | last1 = Balkema | first1 = A.
 |last2=de Haan |first2=Laurens | author-link2 = Laurens de Haan
 | year = 1974
 | title = Residual life time at great age
 | journal = Annals of Probability
 | volume = 2 | issue = 5 | pages = 792–804
 | doi = 10.1214/aop/1176996548 | doi-access = free
 | jstor = 2959306
}}
* {{cite book
 | last = Burry | first = K.V.
 | year = 1975
 | title = Statistical Methods in Applied Science
 | place = Hoboken, NJ
 | publisher = John Wiley & Sons
}}
* {{cite book
 | last = Castillo | first = E.
 | year = 1988
 | title = Extreme Value Theory in Engineering
 | publisher = Academic Press
 | place = New York, NY
 | isbn = 0-12-163475-2
}}
* {{cite book
 | last1 = Castillo     | first1 = E.           | last2 = Hadi     | first2 = A.S.
 | last3 = Balakrishnan | first3 = N.           | last4 = Sarabia  | first4 = J.M.
 | year = 2005
 | title = Extreme Value and Related Models with Applications in Engineering and Science
 | series = Wiley Series in Probability and Statistics 
 | place = Hoboken, NJ
 | publisher = John Wiley's Sons
 | isbn = 0-471-67172-X
}}
* {{cite book
 | last = Coles | first = S.
 | year = 2001
 | title = An Introduction to Statistical Modeling of Extreme Values
 | publisher = Springer
 | place = London, UK
}}
* {{cite book
 | last1 = Embrechts   | first1 = P.
 | last2 = Klüppelberg | first2 = C.  |author2-link=Claudia Klüppelberg 
 | last3 = Mikosch     | first3 = T.
 | year = 1997
 | title = Modelling extremal events for insurance and finance
 | place = Berlin, DE
 | publisher = Springer Verlag
}}
* {{cite journal
 | last1 = Fisher  | first1 = R.A.    | author1-link = Ronald Fisher
 | last2 = Tippett | first2 = L.H.C.  | author2-link = Leonard Tippett
 | year = 1928
 | title = Limiting forms of the frequency distribution of the largest and smallest member of a sample
 | journal = [[Proceedings of the Cambridge Philosophical Society]]
 | volume = 24 | issue = 2 | pages = 180–190
 | doi=10.1017/s0305004100015681
 | bibcode = 1928PCPS...24..180F | s2cid = 123125823 
}}
* {{cite journal
 | last1 = Gnedenko | first1 = B.V.
 | year = 1943
 | title = Sur la distribution limite du terme maximum d'une serie aleatoire | language = fr
 | trans-title = On the limiting distribution(s) of the maximum value of a series ...
 | journal = [[Annals of Mathematics]]
 | volume = 44 | issue = 3 | pages = 423–453
 | doi=10.2307/1968974 | jstor = 1968974
}}
* {{cite journal
 |editor-last=Gumbel  |editor-first=E.J.  |editor-link=Emil Julius Gumbel
 |publication-date=1935  |orig-date=1933–1934
 |title=Les valeurs extrêmes des distributions statistiques |language=fr
 |trans-title=The statistical distributions of extreme values
 |journal=[[Annales de l'Institut Henri Poincaré]] 
 |volume=5  |issue=2  |pages=115–158
 |type=conference papers
 |via=numdam.org
 |place=France
 |url=http://www.numdam.org/item?id=AIHP_1935__5_2_115_0
 |access-date=2009-04-01  |format=pdf 
}}
* {{cite book
 | first=E.J. | last = Gumbel  | author-link = Emil Julius Gumbel
 | orig-date = 1958 | year = 2004
 | title = Statistics of Extremes  |edition=reprint
 | publisher = Dover
 | location = Mineola, NY
 | isbn = 978-0-486-43604-3
 | url = https://books.google.com/books?id=kXCg8B5xSUwC&pg=PP1
}}
* {{cite journal
 | last = Makkonen  | first = L.
 | year = 2008
 | title = Problems in the extreme value analysis
 | journal = Structural Safety
 | volume = 30 | issue = 5 | pages = 405–419
 | doi = 10.1016/j.strusafe.2006.12.001
}}
* {{cite journal
 | last = Leadbetter | first = M.R.
 | year = 1991
 | title = On a basis for 'peaks over threshold' modeling
 | journal = Statistics & Probability Letters
 | volume = 12 | issue = 4 | pages = 357–362
 | doi=10.1016/0167-7152(91)90107-3
}}
*  {{cite book
 | last1 = Leadbetter | first1 = M.R.
 | last2 = Lindgren   | first2 = G.
 | last3 = Rootzen    | first3 = H.
 | year = 1982
 | title = Extremes and Related Properties of Random Sequences and Processes
 | publisher = Springer-Verlag
 | place = New York, NY
}}
* {{cite journal
 | last1 = Lindgren | first1 = G.
 | last2 = Rootzen  | first2 = H.
 | year = 1987
 | title = Extreme values: Theory and technical applications
 | journal = Scandinavian Journal of Statistics, Theory and Applications
 | volume = 14 | pages = 241–279
}}
* {{cite book
 | last = Novak | first = S.Y.
 | year = 2011
 | title = Extreme Value Methods with Applications to Finance
 |publisher=Chapman & Hall / CRC Press
 |place=London, UK / Boca Raton, FL
 |isbn=978-1-4398-3574-6
}} 
* {{cite journal
 | last = Pickands | first = J.
 | year = 1975
 | title = Statistical inference using extreme order statistics
 | journal = Annals of Statistics
 | volume = 3 | pages = 119–131
 | doi = 10.1214/aos/1176343003 | doi-access = free
}}
* {{cite journal
 |last1=Tippett |first1=Michael K.
 |last2=Lepore  |first2=Chiara
 |last3=Cohen   |first3=Joel E.
 |date=16 December 2016
 |title=More tornadoes in the most extreme U.S. tornado outbreaks
 |journal=Science
 |volume=354 |issue=6318 |pages=1419–1423
 |doi=10.1126/science.aah7393 |doi-access=free
 |pmid=27934705 |bibcode=2016Sci...354.1419T
}}
{{refend}}

==Software==

* {{cite journal
 | last1 = Belzile | first1 = L.R.
 | last2 = Dutang | first2 = C. 
 | last3 = Northrop | first3 = P.J. 
 | last4 = Opitz | first4 = T.
 | year = 2023
 | title = A modeler's guide to extreme value software
 | journal = Extremes
 | volume = 26 | issue = 4
 | pages = 595–638
 | doi = 10.1007/s10687-023-00475-9
| arxiv = 2205.07714}}
* {{cite web
 |title=Extreme Value Statistics in R 
 |type=software
 |website=cran.r-project.org
 |date=4 November 2023
 |url=https://cran.r-project.org/web/views/ExtremeValue.html
}} — Package for extreme value statistics in [[R (programming language)|R]].
* {{cite web
 |title=Extremes.jl
 |type=software
 |website=github.com
 |url=https://github.com/juliohm/ExtremeStats.jl
}} — Package for extreme value statistics in [[Julia (programming language)|Julia]].
* {{cite web
 |title=Source code for stationary and non-stationary extreme value analysis
 |type=software
 |website=amir.eng.uci.edu |place=Irvine, CA
 |publisher=[[University of California, Irvine]]
 |url=http://amir.eng.uci.edu/neva.php
}}

==External links==
{{refbegin |colwidth=25em |small=yes}}

* {{cite report
 |last1=Chavez-Demoulin |first1=Valérie
 |last2=Roehrl          |first2=Armin
 |date=8 January 2004
 |title=Extreme value theory can save your neck |lang=en
 |website=risknet.de
 |place=Germany
 |url=http://www.risknet.de/fileadmin/eLibrary/EVT-Paper-Roehrl-Chavez-Demoulin.pdf
}} — Easy non-mathematical introduction.

* {{cite report
 |title=Steps in applying extreme value theory to finance: A review
 |date=c. 2010  |publication-date=January 2010
 |website=bankofcanada.ca
 |publisher=Bank of Canada
 |url=http://www.bankofcanada.ca/wp-content/uploads/2010/01/wp00-20.pdf
}}

* {{cite journal
 |editor-last=Gumbel  |editor-first=E.J.  |editor-link=Emil Julius Gumbel
 |publication-date=1935  |orig-date=1933–1934
 |title=Les valeurs extrêmes des distributions statistiques |language=fr
 |trans-title=The statistical distributions of extreme values
 |journal=[[Annales de l'Institut Henri Poincaré]] 
 |volume=5  |issue=2  |pages=115–158
 |type=conference papers
 |via=numdam.org
 |place=France
 |url=http://www.numdam.org/item?id=AIHP_1935__5_2_115_0
 |access-date=2009-04-01  |format=pdf 
}} — Full-text access to conferences held by {{nobr| [[Emil Julius Gumbel|E.J. Gumbel]] }} in 1933–1934.

{{refend}}

{{Authority control}}

{{DEFAULTSORT:Extreme Value Theory}}
[[Category:Actuarial science]]
[[Category:Statistical theory]]
[[Category:Extreme value data]]
[[Category:Tails of probability distributions]]
[[Category:Financial risk modeling]]