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第1行： − {{too technical|date=May 2020}} − − {{mergefrom|Fat-tailed distribution|date=May 2020}} − − In [[probability theory]], '''heavy-tailed distributions''' are [[probability distribution]]s whose tails are not exponentially bounded:<ref name="Asmussen">{{Cite book | doi = 10.1007/0-387-21525-5_10 | first = S. R. | last = Asmussen| chapter = Steady-State Properties of GI/G/1 | title = Applied Probability and Queues | series = Stochastic Modelling and Applied Probability | volume = 51 | pages = 266–301 | year = 2003 | isbn = 978-0-387-00211-8 | pmid = | pmc = }}</ref> that is, they have heavier tails than the [[exponential distribution]]. In many applications it is the right tail of the distribution that is of interest, but a distribution may have a heavy left tail, or both tails may be heavy. − − In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distribution that is of interest, but a distribution may have a heavy left tail, or both tails may be heavy. − − There are three important subclasses of heavy-tailed distributions: the [[fat-tailed distribution]]s, the [[long-tailed distribution]]s 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. 第21行：
第10行： − 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 [[Moment (mathematics)|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.) 第29行：
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无编辑摘要
此词条Jie翻译。已由Smile审校。
此词条Jie翻译。已由Smile审校。
在概率论中，<font color="#ff8000">重尾分布 Heavy-tailed distributions</font>是指其尾部呈现出不受指数限制的概率分布<ref name="Asmussen">{{Cite book | doi = 10.1007/0-387-21525-5_10 | first = S. R. | last = Asmussen| chapter = Steady-State Properties of GI/G/1 | title = Applied Probability and Queues | series = Stochastic Modelling and Applied Probability | volume = 51 | pages = 266–301 | year = 2003 | isbn = 978-0-387-00211-8 | pmid = | pmc = }}</ref>：也就是说，它们的尾部比<font color="#ff8000">指数分布 exponential distribution </font> “重”。在许多应用中，关注的是分布的右尾，但是分布的左尾可能也很重，或者两个尾都很重。
在概率论中，<font color="#ff8000">重尾分布 Heavy-tailed distributions</font>是指其尾部呈现出不受指数限制的概率分布<ref name="Asmussen">{{Cite book | doi = 10.1007/0-387-21525-5_10 | first = S. R. | last = Asmussen| chapter = Steady-State Properties of GI/G/1 | title = Applied Probability and Queues | series = Stochastic Modelling and Applied Probability | volume = 51 | pages = 266–301 | year = 2003 | isbn = 978-0-387-00211-8 | pmid = | pmc = }}</ref>：也就是说，它们的尾部比<font color="#ff8000">指数分布 exponential distribution </font> “重”。在许多应用中，关注的是分布的右尾，但是分布的左尾可能也很重，或者两个尾都很重。
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<font color="#ff8000">次指数分布类 subexponential class </font>。
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<font color="#ff8000">次指数分布类 subexponential class </font>。
在使用<font color="#ff8000">“重尾” Heavy-tailed</font>一词时仍存在一些歧义。于是就出现了另外两种定义。一些作者使用该术语来指代并非所有幂矩都是有限的那些分布，以及其它一些没有有限方差的分布。本文中给出的是最常用的定义，包括替代定义所涵盖的所有分布，以及具有所有幂矩的<font color="#ff8000">对数正态分布 long-normal distributions </font>，但通常被认为是重尾的。（有时“重尾”用于任何具有比正态分布更重的尾巴的分布。）
在使用<font color="#ff8000">“重尾” Heavy-tailed</font>一词时仍存在一些歧义。于是就出现了另外两种定义。一些作者使用该术语来指代并非所有幂矩都是有限的那些分布，以及其它一些没有有限方差的分布。本文中给出的是最常用的定义，包括替代定义所涵盖的所有分布，以及具有所有幂矩的<font color="#ff8000">对数正态分布 long-normal distributions </font>，但通常被认为是重尾的。（有时“重尾”用于任何具有比正态分布更重的尾巴的分布。）
== Definitions 定义 ==
== 定义 ==
=== Definition of heavy-tailed distribution 重尾分布的定义 ===
=== 重尾分布的定义 ===
The distribution of a [[random variable]] ''X'' with [[cumulative distribution function|distribution function]] ''F'' is said to have a heavy (right) tail if the [[moment generating function]] of ''X'', ''M<sub>X</sub>''(''t''), is infinite for all ''t'' > 0.<ref name="ReferenceA">Rolski, Schmidli, Scmidt, Teugels, ''Stochastic Processes for Insurance and Finance'', 1999</ref>
The distribution of a [[random variable]] ''X'' with [[cumulative distribution function|distribution function]] ''F'' is said to have a heavy (right) tail if the [[moment generating function]] of ''X'', ''M<sub>X</sub>''(''t''), is infinite for all ''t'' > 0.<ref name="ReferenceA">Rolski, Schmidli, Scmidt, Teugels, ''Stochastic Processes for Insurance and Finance'', 1999</ref>
The distribution of a random variable X with distribution function F is said to have a heavy (right) tail if the moment generating function of X, MX(t), is infinite for all t > 0.
The distribution of a random variable X with distribution function F is said to have a heavy (right) tail if the moment generating function of X, MX(t), is infinite for all t > 0.
如果''X''的矩生成函数, ''M<sub>X</sub>''(''t'')对于所有''t'' > 0都是无限的，则具有分布函数''F''的随机变量''X''的分布被称为重尾（右）。<ref name="ReferenceA">Rolski, Schmidli, Scmidt, Teugels, ''Stochastic Processes for Insurance and Finance'', 1999</ref>
如果''<math>X</math>''的矩生成函数, ''M<sub>X</sub>''(''t'')对于所有''t'' > 0都是无限的，则具有分布函数''F''的随机变量''X''的分布被称为重尾（右）。<ref name="ReferenceA">Rolski, Schmidli, Scmidt, Teugels, ''Stochastic Processes for Insurance and Finance'', 1999</ref>