更改

There are three important subclasses of heavy-tailed distributions: the fat-tailed distributions, the long-tailed distributions and the subexponential distributions.  In practice, all commonly used heavy-tailed distributions belong to the subexponential class.

There are three important subclasses of heavy-tailed distributions: the fat-tailed distributions, the long-tailed distributions and the subexponential distributions.  In practice, all commonly used heavy-tailed distributions belong to the subexponential class.

重尾分布有三个重要的子类：'''<font color="#ff8000"> 文字</font>'''胖尾分布Fat-tailed distribution，'''<font color="#ff8000"> 长尾分布Long-tailed distribution</font>'''和'''<font color="#ff8000"> 次指数分布Subexponential distributions</font>'''。实际上，所有常用的重尾分布都属于次指数类分布。

重尾分布有三个重要的子类：'''<font color="#ff8000"> 胖尾分布Fat-tailed distribution</font>'''，'''<font color="#ff8000"> 长尾分布Long-tailed distribution</font>'''和'''<font color="#ff8000"> 次指数分布Subexponential distributions</font>'''。实际上，所有常用的重尾分布都属于次指数类分布。

There is still some discrepancy over the use of the term heavy-tailed.  There are two other definitions in use.  Some authors  use the term to refer to those distributions which do not have all their power moments finite; and some others to those distributions that do not have a finite variance.  The definition given in this article is the most general in use, and includes all distributions encompassed by the alternative definitions, as well as those distributions such as log-normal that possess all their power moments, yet which are generally considered to be heavy-tailed.  (Occasionally, heavy-tailed is used for any distribution that has heavier tails than the normal distribution.)

There is still some discrepancy over the use of the term heavy-tailed.  There are two other definitions in use.  Some authors  use the term to refer to those distributions which do not have all their power moments finite; and some others to those distributions that do not have a finite variance.  The definition given in this article is the most general in use, and includes all distributions encompassed by the alternative definitions, as well as those distributions such as log-normal that possess all their power moments, yet which are generally considered to be heavy-tailed.  (Occasionally, heavy-tailed is used for any distribution that has heavier tails than the normal distribution.)

Those that are one-tailed include:

Those that are one-tailed include:

单尾的包括：

单尾的包括：

* 帕累托分布Pareto distribution;

* '''<font color="#ff8000"> 帕累托分布Pareto distribution</font>''';

* 对数正态分布Log-normal distribution;

* '''<font color="#ff8000"> 对数正态分布Log-normal distribution</font>''';

* 莱维分布Lévy distribution;

* '''<font color="#ff8000"> 莱维分布Lévy distribution</font>''';

* 形状参数大于0但小于1的韦布尔分布Weibull distribution

* 形状参数大于0但小于1的'''<font color="#ff8000"> 韦布尔分布Weibull distribution</font>''';

* 伯尔分布Burr distribution;

* '''<font color="#ff8000"> 伯尔分布Burr distribution</font>''';

* 对数逻辑分布log-logistic distribution;

* '''<font color="#ff8000"> 对数逻辑分布log-logistic distribution</font>''';

* 对数伽玛分布log-gamma distribution;

* '''<font color="#ff8000"> 对数伽玛分布log-gamma distribution</font>''';

* 弗雷歇分布Fréchet distribution;

* '''<font color="#ff8000"> 弗雷歇分布Fréchet distribution</font>''';

* 对数柯西分布log-Cauchy distribution，有时被描述为“超重尾”分布，因为它表现出对数衰减，从而产生比帕累托分布更重的尾。

* '''<font color="#ff8000"> 对数柯西分布log-Cauchy distribution</font>'''，有时被描述为“超重尾”分布，因为它表现出对数衰减，从而产生比帕累托分布更重的尾。

*  

*  

Those that are two-tailed include:

Those that are two-tailed include:

双尾的包括：

双尾的包括：

* 柯西分布Cauchy distribution本身就是稳定分布和t分布的特例；

* '''<font color="#ff8000"> 柯西分布Cauchy distribution</font>'''本身就是稳定分布和t分布的特例；

* 稳定分布族The family of stable distributions，但该族中正态分布的特殊情况除外。一些稳定的分布是单面的（或由半线的），例如莱维分布Lévy distribution。另请参见具有长尾分布和波动性聚类的财务模型。

* '''<font color="#ff8000"> 稳定分布族The family of stable distributions</font>'''，但该族中正态分布的特殊情况除外。一些稳定的分布是单面的（或由半线的），例如'''<font color="#ff8000"> 莱维分布Lévy distribution</font>'''。另请参见具有长尾分布和波动性聚类的财务模型。

* t分布

* t分布

* 偏对数正态级联分布。

* 偏对数正态级联分布。

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

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

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}} 尾指数估计 ==

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

* 尖峭态分布Leptokurtic distribution

* '''<font color="#ff8000"> 尖峭态分布Leptokurtic distribution</font>'''

* 广义极值分布Generalized extreme value distribution

* '''<font color="#ff8000"> 广义极值分布Generalized extreme value distribution</font>'''

* 离群值Outlier

* '''<font color="#ff8000"> 离群值Outlier</font>'''

* 长尾Long tail

* '''<font color="#ff8000"> 长尾Long tail</font>'''

* 幂律Power law

* '''<font color="#ff8000"> 幂律Power law</font>'''

* 随机的七个状态Seven states of randomness

* '''<font color="#ff8000"> 随机的七个状态Seven states of randomness</font>'''

* 胖尾分布Fat-tailed distribution

* '''<font color="#ff8000"> 胖尾分布Fat-tailed distribution</font>'''

**塔勒布分布Taleb distribution和圣杯分布Holy grail distribution

**'''<font color="#ff8000"> 塔勒布分布Taleb distribution</font>'''和'''<font color="#ff8000"> 圣杯分布Holy grail distribution</font>'''

 

 

==References==

== References 参考文献 ==

<references/>

<references/>