重尾分布

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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.

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.

在概率论中，重尾分布是指其尾部呈现出不受指数限制的概率分布：也就是说，它们的尾部比指数分布“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。

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：肥尾分布，长尾分布和次指数分布。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.)

在使用“重尾”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那些并非所有幂矩都是有限的分布。也有其它一些人以此指代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对数正态分布，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尾巴的分布。）

Definitions 定义

Definition of heavy-tailed distribution 重尾分布的定义

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, M_X(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_X(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）。

That means

也就是说

[math]\displaystyle{ \int_{-\infty}^\infty e^{t x} \,dF(x) = \infty \quad \mbox{for all } t\gt 0. }[/math]

An implication of this is that

这意味着

[math]\displaystyle{ \lim_{x \to \infty} e^{t x}\Pr[X\gt x] = \infty \quad \mbox{for all } t\gt 0.\, }[/math]

This is also written in terms of the tail distribution function

[math]\displaystyle{ \overline{F}(x) ≡ \Pr[X\gt x] }[/math]

as

[math]\displaystyle{ \lim_{x \to \infty} e^{t x}\overline{F}(x) = \infty \quad \mbox{for all } t \gt 0.\, }[/math]

Definition of long-tailed distribution 长尾分布的定义

The distribution of a random variable X with distribution function F is said to have a long right tail if for all t > 0,

The distribution of a random variable X with distribution function F is said to have a long right tail[1] if for all t > 0,

如果对于所有t>0，具有分布函数F的随机变量X的分布具有较长的右尾，

[math]\displaystyle{ \lim_{x \to \infty} \Pr[X\gt x+t\mid X\gt x] =1, \, }[/math]

or equivalently 或等同于

[math]\displaystyle{ \overline{F}(x+t) \sim \overline{F}(x) \quad \mbox{as } x \to \infty. \, }[/math]

This has the intuitive interpretation for a right-tailed long-tailed distributed quantity that if the long-tailed quantity exceeds some high level, the probability approaches 1 that it will exceed any other higher level.

This has the intuitive interpretation for a right-tailed long-tailed distributed quantity that if the long-tailed quantity exceeds some high level, the probability approaches 1 that it will exceed any other higher level.

对于右尾长尾分布量，该解释非常直观：即如果长尾量超过某个高水平，则概率将接近1，它将超过任何其他更高水平。

All long-tailed distributions are heavy-tailed, but the converse is false, and it is possible to construct heavy-tailed distributions that are not long-tailed.

All long-tailed distributions are heavy-tailed, but the converse is false, and it is possible to construct heavy-tailed distributions that are not long-tailed.

所有长尾分布都是重尾分布，但反之不一定，事实是可以构造出非长尾分布的重尾分布。

Subexponential distributions 长尾分布的定义

Subexponentiality is defined in terms of convolutions of probability distributions. For two independent, identically distributed random variables [math]\displaystyle{ X_1,X_2 }[/math] with common distribution function [math]\displaystyle{ F }[/math] the convolution of [math]\displaystyle{ F }[/math] with itself, [math]\displaystyle{ F^{*2} }[/math] is convolution square, using Lebesgue–Stieltjes integration, by:

Subexponentiality is defined in terms of convolutions of probability distributions. For two independent, identically distributed random variables [math]\displaystyle{ X_1,X_2 }[/math] with common distribution function [math]\displaystyle{ F }[/math] the convolution of [math]\displaystyle{ F }[/math] with itself, [math]\displaystyle{ F^{*2} }[/math] is convolution square, using Lebesgue–Stieltjes integration, by:

次指数性是根据概率分布的卷积定义的。对于具有共同分布函数F的两个独立的，分布均匀的随机变量X1，X2，F与自身的卷积，F2是卷积平方，使用Lebesgue–Stieltjes积分，方法如下：

[math]\displaystyle{ \Pr[X_1+X_2 \leq x] = F^{*2}(x) = \int_{0}^x F(x-y)\,dF(y), }[/math]

and the n-fold convolution [math]\displaystyle{ F^{*n} }[/math] is defined inductively by the rule:

n倍卷积[math]\displaystyle{ F^{*n} }[/math]定义如下：

[math]\displaystyle{ F^{*n}(x) = \int_{0}^x F(x-y)\,dF^{*n-1}(y). }[/math]

The tail distribution function [math]\displaystyle{ \overline{F} }[/math] is defined as [math]\displaystyle{ \overline{F}(x) = 1-F(x) }[/math].

尾分布函数[math]\displaystyle{ \overline{F} }[/math]定义为[math]\displaystyle{ \overline{F}(x) = 1-F(x) }[/math]。

A distribution [math]\displaystyle{ F }[/math] on the positive half-line is subexponential if

如果满足以下条件，则正半线上的分布[math]\displaystyle{ F }[/math]为次指数：

[math]\displaystyle{ \overline{F^{*2}}(x) \sim 2\overline{F}(x) \quad \mbox{as } x \to \infty. }[/math]

This implies that, for any [math]\displaystyle{ n \geq 1 }[/math],

这蕴含着，对于任何[math]\displaystyle{ n \geq 1 }[/math]，

[math]\displaystyle{ \overline{F^{*n}}(x) \sim n\overline{F}(x) \quad \mbox{as } x \to \infty. }[/math]

The probabilistic interpretation of this is that, for a sum of [math]\displaystyle{ n }[/math] independent random variables [math]\displaystyle{ X_1,\ldots,X_n }[/math] with common distribution [math]\displaystyle{ F }[/math],

[math]\displaystyle{ \Pr[X_1+ \cdots +X_n\gt x] \sim \Pr[\max(X_1, \ldots,X_n)\gt x] \quad \text{as } x \to \infty. }[/math]

This is often known as the principle of the single big jump or catastrophe principle.

A distribution [math]\displaystyle{ F }[/math] on the whole real line is subexponential if the distribution [math]\displaystyle{ F I([0,\infty)) }[/math] is. Here [math]\displaystyle{ I([0,\infty)) }[/math] is the indicator function of the positive half-line. Alternatively, a random variable [math]\displaystyle{ X }[/math] supported on the real line is subexponential if and only if [math]\displaystyle{ X^+ = \max(0,X) }[/math] is subexponential.

All subexponential distributions are long-tailed, but examples can be constructed of long-tailed distributions that are not subexponential.

Common heavy-tailed distributions

All commonly used heavy-tailed distributions are subexponential.

Those that are one-tailed include:

• the Pareto distribution;

• the Log-normal distribution;

• the Lévy distribution;

• the Weibull distribution with shape parameter greater than 0 but less than 1;

Category:Tails of probability distributions

类别: 概率分布的尾部

• the Burr distribution;

Category:Types of probability distributions

类别: 概率分布的类型

• the log-logistic distribution;

Category:Actuarial science

类别: 精算

• the log-gamma distribution;

Category:Risk

类别: 风险

This page was moved from wikipedia:en:Heavy-tailed distribution. Its edit history can be viewed at 重尾分布/edithistory