Editing Extreme value theory (section)

==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>