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添加162字节 、 2020年10月18日 (日) 16:51
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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。
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==Common heavy-tailed distributions==
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== 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"/>
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* 偏对数正态级联分布。
 
* 偏对数正态级联分布。
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== Relationship to fat-tailed distributions ==
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== 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。
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== Estimating the tail-index{{definition|date=January 2018}} ==
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== 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
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=== Hill's tail-index estimator ===
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=== 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  
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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。
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=== Ratio estimator of the tail-index ===
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=== 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   
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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"/>
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===Software===
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=== 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>
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==Estimation of heavy-tailed density==
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== 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  
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