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→Estimating the tail-index{{definition|date=January 2018}}
| isbn=978-1-43983-574-6
| isbn=978-1-43983-574-6
}}</ref>) approaches to the problem of the tail-index estimation.
}}</ref>) approaches to the problem of the tail-index estimation.
对于尾指数估计的问题，有参数方法（参见Emprechts等人）和非参数方法（例如，Novak）两种。
To estimate the tail-index using the parametric approach, some authors employ [[GEV distribution]] or [[Pareto distribution]]; they may apply the maximum-likelihood estimator (MLE).
To estimate the tail-index using the parametric approach, some authors employ [[GEV distribution]] or [[Pareto distribution]]; they may apply the maximum-likelihood estimator (MLE).
=== Pickand's tail-index estimator ===
为了使用参数化方法估计尾指数，有些作者采用了GEV分布或帕累托分布；他们可能会运用最大似然估计器（MLE）。
=== Pickand's tail-index estimator Pickand的尾指数估算器===
With <math>(X_n , n \geq 1)</math> a random sequence of independent and same density function <math>F \in D(H(\xi))</math>, the Maximum Attraction Domain<ref name=Pickands>{{cite journal|last=Pickands III|first=James|title=Statistical Inference Using Extreme Order Statistics|journal=The Annals of Statistics|date=Jan 1975|volume=3|issue=1|pages=119–131|jstor=2958083|doi=10.1214/aos/1176343003|doi-access=free}}</ref> of the generalized extreme value density <math> H </math>, where <math>\xi \in \mathbb{R}</math>. If <math>\lim_{n\to\infty} k(n) = \infty </math> and <math>\lim_{n\to\infty} \frac{k(n)}{n}= 0</math>, then the ''Pickands'' tail-index estimation is<ref name="Embrechts"/><ref name="Pickands"/>
With <math>(X_n , n \geq 1)</math> a random sequence of independent and same density function <math>F \in D(H(\xi))</math>, the Maximum Attraction Domain<ref name=Pickands>{{cite journal|last=Pickands III|first=James|title=Statistical Inference Using Extreme Order Statistics|journal=The Annals of Statistics|date=Jan 1975|volume=3|issue=1|pages=119–131|jstor=2958083|doi=10.1214/aos/1176343003|doi-access=free}}</ref> of the generalized extreme value density <math> H </math>, where <math>\xi \in \mathbb{R}</math>. If <math>\lim_{n\to\infty} k(n) = \infty </math> and <math>\lim_{n\to\infty} \frac{k(n)}{n}= 0</math>, then the ''Pickands'' tail-index estimation is<ref name="Embrechts"/><ref name="Pickands"/>
===Software===
===Software===
* [http://www.cs.bu.edu/~crovella/aest.html aest], [[C (programming language)|C]] tool for estimating the heavy-tail index.<ref>{{Cite journal | last1 = Crovella | first1 = M. E. | last2 = Taqqu | first2 = M. S. | title = Estimating the Heavy Tail Index from Scaling Properties| journal = Methodology and Computing in Applied Probability | volume = 1 | pages = 55–79 | year = 1999 | doi = 10.1023/A:1010012224103 | url = http://www.cs.bu.edu/~crovella/paper-archive/aest.ps| pmid = | pmc = }}</ref>
* [http://www.cs.bu.edu/~crovella/aest.html aest], [[C (programming language)|C]] tool for estimating the heavy-tail index.<ref>{{Cite journal | last1 = Crovella | first1 = M. E. | last2 = Taqqu | first2 = M. S. | title = Estimating the Heavy Tail Index from Scaling Properties| journal = Methodology and Computing in Applied Probability | volume = 1 | pages = 55–79 | year = 1999 | doi = 10.1023/A:1010012224103 | url = http://www.cs.bu.edu/~crovella/paper-archive/aest.ps| pmid = | pmc = }}</ref>
==Estimation of heavy-tailed density==
==Estimation of heavy-tailed density==