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第165行：
第165行： − + − There are parametric (see Embrechts et al.<ref name="Embrechts"/>) and non-parametric (see, e.g., Novak<ref name="Novak2011">{{cite book − | author=Novak S.Y. − | title=Extreme value methods with applications to finance − | year=2011 − | series=London: CRC − | isbn=978-1-43983-574-6 − }}</ref>) approaches to the problem of the tail-index estimation. 第184行：
第177行： − − 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). 第191行：
第182行： − + − 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"/> 第203行：
第193行： − where <math>X_{(n-k(n)+1,n)}=\max \left(X_{n-k(n)+1},\ldots ,X_{n}\right)</math>. This estimator converges in probability to <math>\xi</math>. 第209行：
第198行： − + − Let <math>(X_t , t \geq 1)</math> be a sequence of independent and identically distributed random variables with distribution function <math>F \in D(H(\xi))</math>, the maximum domain of attraction of the [[generalized extreme value distribution]] <math> H </math>, where <math>\xi \in \mathbb{R}</math>. The sample path is <math>{X_t: 1 \leq t \leq n}</math> where <math>n</math> is the sample size. If − <math>\{k(n)\}</math> is an intermediate order sequence, i.e. <math>k(n) \in \{1,\ldots,n-1\}, </math>, <math>k(n) \to \infty</math> and <math>k(n)/n \to 0</math>, then the Hill tail-index estimator is<ref>Hill B.M. (1975) A simple general approach to inference about the tail of a distribution. Ann. Stat., v. 3, 1163–1174.</ref> 第222行：
第209行： − where <math>X_{(i,n)}</math> is the <math>i</math>-th [[order statistic]] of <math>X_1, \dots, X_n</math>. − This estimator converges in probability to <math>\xi</math>, and is asymptotically normal provided <math>k(n) \to \infty </math> is restricted based on a higher order regular variation property<ref>Hall, P.(1982) On some estimates of an exponent of regular variation. J. R. Stat. Soc. Ser. B., v. 44, 37–42.</ref> − .<ref>Haeusler, E. and J. L. Teugels (1985) On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann. Stat., v. 13, 743–756.</ref> Consistency and asymptotic normality extend to a large class of dependent and heterogeneous sequences,<ref>Hsing, T. (1991) On tail index estimation using dependent data. Ann. Stat., v. 19, 1547–1569.</ref><ref>Hill, J. (2010) On tail index estimation for dependent, heterogeneous data. Econometric Th., v. 26, 1398–1436.</ref> irrespective of whether <math>X_t</math> is observed, or a computed residual or filtered data from a large class of models and estimators, including mis-specified models and models with errors that are dependent.<ref>Resnick, S. and Starica, C. (1997). Asymptotic behavior of Hill’s estimator for autoregressive data. Comm. Statist. Stochastic Models 13, 703–721.</ref><ref>Ling, S. and Peng, L. (2004). Hill’s estimator for the tail index of an ARMA model. J. Statist. Plann. Inference 123, 279–293.</ref><ref>Hill, J. B. (2015). Tail index estimation for a filtered dependent time series. Stat. Sin. 25, 609–630.</ref> 第230行：
第214行： − + − The ratio estimator (RE-estimator) of the tail-index was introduced by Goldie − and Smith.<ref>Goldie C.M., Smith R.L. (1987) Slow variation with remainder: − theory and applications. Quart. J. Math. Oxford, v. 38, 45–71.</ref> − It is constructed similarly to Hill's estimator but uses a non-random "tuning parameter". 第242行：
第222行： − A comparison of Hill-type and RE-type estimators can be found in Novak.<ref name="Novak2011"/> − + − * [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>
→Estimating the tail-index{{definition|date=January 2018}} 尾指数估计
胖尾分布是这样的分布，对于较大的x，概率密度函数为<math>x^{-a}</math>趋于零。由于这样的幂总是受到指数分布概率密度函数的限制，因此，胖尾分布始终是重尾分布。但是，某些分布的尾部趋近于零的速率比指数函数慢（表示它们是重尾），而比幂快（表示它们不是胖尾）。例如对数正态分布<ref>{{Contradict-inline|article=fat-tailed distribution|reason=Fat-tailed page says log-normals are in fact fat-tailed.|date=June 2019}}</ref>。当然，许多其他的重尾分布，例如对数逻辑分布和帕累托分布也属于胖尾分布。
胖尾分布是这样的分布，对于较大的x，概率密度函数为<math>x^{-a}</math>趋于零。由于这样的幂总是受到指数分布概率密度函数的限制，因此，胖尾分布始终是重尾分布。但是，某些分布的尾部趋近于零的速率比指数函数慢（表示它们是重尾），而比幂快（表示它们不是胖尾）。例如对数正态分布<ref>{{Contradict-inline|article=fat-tailed distribution|reason=Fat-tailed page says log-normals are in fact fat-tailed.|date=June 2019}}</ref>。当然，许多其他的重尾分布，例如对数逻辑分布和帕累托分布也属于胖尾分布。
== Estimating the tail-index{{definition|date=January 2018}} 尾指数估计 ==
== 尾指数估计 ==
对于尾指数估计的问题，有参数方法（参见Emprechts等人<ref name="Embrechts"/>）和非参数方法（例如，Novak<ref name="Novak2011">{{cite book
对于尾指数估计的问题，有参数方法（参见Emprechts等人<ref name="Embrechts"/>）和非参数方法（例如，Novak<ref name="Novak2011">{{cite book
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；他们可能会运用极大似然估计方法（MLE）。
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；他们可能会运用极大似然估计方法（MLE）。
=== Pickand's tail-index estimator Pickand的尾指数估算器===
=== Pickand的尾指数估算器===
对于<math>(X_n , n \geq 1)</math>的独立且相同的密度函数<math>F \in D(H(\xi))</math>的随机序列，是<font color="#ff8000">广义极值密度 the generalized extreme value density </font><math>H</math>的<font color="#ff8000">最大吸引域 the Maximum Attraction Domain </font><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>，其中<math>\xi \in \mathbb{R}</math>。如果<math>\lim_{n\to\infty} k(n) = \infty </math>和<math>\lim_{n\to\infty} \frac{k(n)}{n}= 0</math>，则Pickands尾部指数估计为<ref name="Embrechts"/><ref name="Pickands"/>
对于<math>(X_n , n \geq 1)</math>的独立且相同的密度函数<math>F \in D(H(\xi))</math>的随机序列，是<font color="#ff8000">广义极值密度 the generalized extreme value density </font><math>H</math>的<font color="#ff8000">最大吸引域 the Maximum Attraction Domain </font><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>，其中<math>\xi \in \mathbb{R}</math>。如果<math>\lim_{n\to\infty} k(n) = \infty </math>和<math>\lim_{n\to\infty} \frac{k(n)}{n}= 0</math>，则Pickands尾部指数估计为<ref name="Embrechts"/><ref name="Pickands"/>
其中<math>X_{(n-k(n)+1,n)}=\max \left(X_{n-k(n)+1},\ldots ,X_{n}\right)</math>。 此估计量的概率收敛到<math>\xi</math>。
其中<math>X_{(n-k(n)+1,n)}=\max \left(X_{n-k(n)+1},\ldots ,X_{n}\right)</math>。 此估计量的概率收敛到<math>\xi</math>。
=== Hill's tail-index estimator 希尔 Hill的尾指数估算器 ===
=== Hill的尾指数估算器 ===
令<math>(X_t , t \geq 1)</math>为具有分布函数<math>F \in D(H(\xi))</math>独立且均匀分布的随机变量序列，其分布函数为广义极值分布<math> H </math>的最大吸引域，其中<math>\xi \in \mathbb{R}</math>。样本路径为<math>{X_t: 1 \leq t \leq n}</math>，其中<math>n</math>为样本大小。 如果<math>\{k(n)\}</math>是中间阶数序列，即<math>k(n) \in \{1,\ldots,n-1\}, </math>，<math>k(n) \to \infty</math>和<math>k(n)/n \to 0</math>，则Hill尾指数估计器为<ref>Hill B.M. (1975) A simple general approach to inference about the tail of a distribution. Ann. Stat., v. 3, 1163–1174.</ref>：
令<math>(X_t , t \geq 1)</math>为具有分布函数<math>F \in D(H(\xi))</math>独立且均匀分布的随机变量序列，其分布函数为广义极值分布<math> H </math>的最大吸引域，其中<math>\xi \in \mathbb{R}</math>。样本路径为<math>{X_t: 1 \leq t \leq n}</math>，其中<math>n</math>为样本大小。 如果<math>\{k(n)\}</math>是中间阶数序列，即<math>k(n) \in \{1,\ldots,n-1\}, </math>，<math>k(n) \to \infty</math>和<math>k(n)/n \to 0</math>，则Hill尾指数估计器为<ref>Hill B.M. (1975) A simple general approach to inference about the tail of a distribution. Ann. Stat., v. 3, 1163–1174.</ref>：
其中<math>X_{(i,n)}</math>是<math>X_1, \dots, X_n</math>的第<math>i</math>次序统计量。该估计量依概率收敛于<math>\xi</math>，并且在基于高阶的正则变化性质的情况下，是限制<math>k(n) \to \infty </math>的渐近正态<ref>Hall, P.(1982) On some estimates of an exponent of regular variation. J. R. Stat. Soc. Ser. B., v. 44, 37–42.</ref>.<ref>Haeusler, E. and J. L. Teugels (1985) On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann. Stat., v. 13, 743–756.</ref>。一致性和渐近正态性适用于一大类相关序列和异类序列<ref>Hsing, T. (1991) On tail index estimation using dependent data. Ann. Stat., v. 19, 1547–1569.</ref><ref>Hill, J. (2010) On tail index estimation for dependent, heterogeneous data. Econometric Th., v. 26, 1398–1436.</ref>，而不管是否观测到<math>X_t</math>，或者来自大量模型和估计量（包括错误指定的模型和具有相关误差的模型）计算出的残差或筛选数据。<ref>Resnick, S. and Starica, C. (1997). Asymptotic behavior of Hill’s estimator for autoregressive data. Comm. Statist. Stochastic Models 13, 703–721.</ref><ref>Ling, S. and Peng, L. (2004). Hill’s estimator for the tail index of an ARMA model. J. Statist. Plann. Inference 123, 279–293.</ref><ref>Hill, J. B. (2015). Tail index estimation for a filtered dependent time series. Stat. Sin. 25, 609–630.</ref>
其中<math>X_{(i,n)}</math>是<math>X_1, \dots, X_n</math>的第<math>i</math>次序统计量。该估计量依概率收敛于<math>\xi</math>，并且在基于高阶的正则变化性质的情况下，是限制<math>k(n) \to \infty </math>的渐近正态<ref>Hall, P.(1982) On some estimates of an exponent of regular variation. J. R. Stat. Soc. Ser. B., v. 44, 37–42.</ref>.<ref>Haeusler, E. and J. L. Teugels (1985) On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann. Stat., v. 13, 743–756.</ref>。一致性和渐近正态性适用于一大类相关序列和异类序列<ref>Hsing, T. (1991) On tail index estimation using dependent data. Ann. Stat., v. 19, 1547–1569.</ref><ref>Hill, J. (2010) On tail index estimation for dependent, heterogeneous data. Econometric Th., v. 26, 1398–1436.</ref>，而不管是否观测到<math>X_t</math>，或者来自大量模型和估计量（包括错误指定的模型和具有相关误差的模型）计算出的残差或筛选数据。<ref>Resnick, S. and Starica, C. (1997). Asymptotic behavior of Hill’s estimator for autoregressive data. Comm. Statist. Stochastic Models 13, 703–721.</ref><ref>Ling, S. and Peng, L. (2004). Hill’s estimator for the tail index of an ARMA model. J. Statist. Plann. Inference 123, 279–293.</ref><ref>Hill, J. B. (2015). Tail index estimation for a filtered dependent time series. Stat. Sin. 25, 609–630.</ref>
=== Ratio estimator of the tail-index 尾部指数的比率估计器 ===
===尾部指数的比率估计器 ===
尾指数的比率估计器（RE估计器）由Goldie和Smith提出<ref>Goldie C.M., Smith R.L. (1987) Slow variation with remainder:
尾指数的比率估计器（RE估计器）由Goldie和Smith提出<ref>Goldie C.M., Smith R.L. (1987) Slow variation with remainder:
在Novak中可以找到Hill型和RE型估计量的比较。<ref name="Novak2011"/>
在Novak中可以找到Hill型和RE型估计量的比较。<ref name="Novak2011"/>
=== Software 应用软件===
===应用软件===
* 用于估计重尾指数的软件[http://www.cs.bu.edu/~crovella/aest.html aest]和C。<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>
* 用于估计重尾指数的软件[http://www.cs.bu.edu/~crovella/aest.html aest]和C。<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>