更改

所有次指数分布都是长尾分布，但可以构造非次指数分布的长尾分布示例。

所有次指数分布都是长尾分布，但可以构造非次指数分布的长尾分布示例。

==Common heavy-tailed distributions==

== Common heavy-tailed distributions 常见的重尾分布 ==

All commonly used heavy-tailed distributions are subexponential.<ref name="Embrechts"/>

All commonly used heavy-tailed distributions are subexponential.<ref name="Embrechts"/>

* 偏对数正态级联分布。

* 偏对数正态级联分布。

== Relationship to fat-tailed distributions ==

== Relationship to fat-tailed distributions 与胖尾分布的关系 ==

A [[fat-tailed distribution]] is a distribution for which the probability density function, for large x, goes to zero as a power <math>x^{-a}</math>.  Since such a power is always bounded below by the probability density function of an exponential distribution, fat-tailed distributions are always heavy-tailed.  Some distributions, however, have a tail which goes to zero slower than an exponential function (meaning they are heavy-tailed), but faster than a power (meaning they are not fat-tailed). An example is the [[log-normal distribution]] {{Contradict-inline|article=fat-tailed distribution|reason=Fat-tailed page says log-normals are in fact fat-tailed.|date=June 2019}}.  Many other heavy-tailed distributions such as the [[log-logistic distribution|log-logistic]] and [[Pareto distribution|Pareto]] distribution are, however, also fat-tailed.

A [[fat-tailed distribution]] is a distribution for which the probability density function, for large x, goes to zero as a power <math>x^{-a}</math>.  Since such a power is always bounded below by the probability density function of an exponential distribution, fat-tailed distributions are always heavy-tailed.  Some distributions, however, have a tail which goes to zero slower than an exponential function (meaning they are heavy-tailed), but faster than a power (meaning they are not fat-tailed). An example is the [[log-normal distribution]] {{Contradict-inline|article=fat-tailed distribution|reason=Fat-tailed page says log-normals are in fact fat-tailed.|date=June 2019}}.  Many other heavy-tailed distributions such as the [[log-logistic distribution|log-logistic]] and [[Pareto distribution|Pareto]] distribution are, however, also fat-tailed.

胖尾分布是这样的分布：对于大x，概率密度函数作为幂<math>x^{-a}</math>变为零。由于幂总是受到指数分布的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。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：对于大x，概率密度函数作为幂<math>x^{-a}</math>变为零。由于幂总是受到指数分布的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。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}} ==

== Estimating the tail-index{{definition|date=January 2018}} 尾指数估计 ==

There are parametric (see Embrechts et al.<ref name="Embrechts"/>) and non-parametric (see, e.g., Novak<ref name="Novak2011">{{cite book

There are parametric (see Embrechts et al.<ref name="Embrechts"/>) and non-parametric (see, e.g., Novak<ref name="Novak2011">{{cite book

=== Hill's tail-index estimator ===

=== Hill's tail-index estimator 希尔的尾指数估算器 ===

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  

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  

其中<math>X_{(i,n)}</math>是<math>X_1, \dots, X_n</math>的i阶统计量。该估计量收敛于<math>\xi</math>的概率，并且当<math>k(n) \to \infty  </math>基于较高阶的正则变化性质受到限制时，它是渐近正态的。一致性和渐近正态性适用于一大类相关序列和异类序列，它与是否观测到<math>X_t</math>无关，也无关于是否从大量模型和估计量（包括错误指定的模型和具有相关误差的模型）中计算出的残差或滤波数据。

其中<math>X_{(i,n)}</math>是<math>X_1, \dots, X_n</math>的i阶统计量。该估计量收敛于<math>\xi</math>的概率，并且当<math>k(n) \to \infty  </math>基于较高阶的正则变化性质受到限制时，它是渐近正态的。一致性和渐近正态性适用于一大类相关序列和异类序列，它与是否观测到<math>X_t</math>无关，也无关于是否从大量模型和估计量（包括错误指定的模型和具有相关误差的模型）中计算出的残差或滤波数据。

=== Ratio estimator of the tail-index ===

=== Ratio estimator of the tail-index 尾部指数的比率估计器 ===

The ratio estimator (RE-estimator) of the tail-index was introduced by Goldie   

The ratio estimator (RE-estimator) of the tail-index was introduced by Goldie   

A comparison of Hill-type and RE-type estimators can be found in Novak.<ref name="Novak2011"/>

A comparison of Hill-type and RE-type estimators can be found in Novak.<ref name="Novak2011"/>

===Software===

=== Software 应用软件===

* [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 重尾密度的估计 ==

Nonparametric approaches to estimate heavy- and superheavy-tailed probability density functions were given in  

Nonparametric approaches to estimate heavy- and superheavy-tailed probability density functions were given in