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 次指数分布

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],

这意味着，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.

这通常被称为单跳或巨灾原理。

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.

如果分布[math]\displaystyle{ F I([0,\infty)) }[/math]为实数，则整个实线上的分布[math]\displaystyle{ F }[/math]是次指数的。此时[math]\displaystyle{ I([0,\infty)) }[/math]是正半线的指标函数。 又或者，当且仅当[math]\displaystyle{ X^+ = \max(0,X) }[/math]是次指数时，实线上支持的随机变量[math]\displaystyle{ X }[/math]才是次指数。

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

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

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;
• the Burr distribution;
• the log-logistic distribution;
• the log-gamma distribution;
• the Fréchet distribution;
• the log-Cauchy distribution, sometimes described as having a "super-heavy tail" because it exhibits logarithmic decay producing a heavier tail than the Pareto distribution.

Those that are two-tailed include:

• The Cauchy distribution, itself a special case of both the stable distribution and the t-distribution;
• The family of stable distributions, excepting the special case of the normal distribution within that family. Some stable distributions are one-sided (or supported by a half-line), see e.g. Lévy distribution. See also financial models with long-tailed distributions and volatility clustering.
• The t-distribution.
• The skew lognormal cascade distribution.

All commonly used heavy-tailed distributions are subexponential.[6] 所有常用的重尾分布都是次指数的。

Those that are one-tailed include: 单尾的包括：

• 帕累托分布Pareto distribution;
• 对数正态分布Log-normal distribution;
• 莱维分布Lévy distribution;
• 形状参数大于0但小于1的韦布尔分布Weibull distribution
• 伯尔分布Burr distribution;
• 对数逻辑分布log-logistic distribution;
• 对数伽玛分布log-gamma distribution;
• 弗雷歇分布Fréchet distribution;
• 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。

Those that are two-tailed include: 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；
• 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。
• t分布
• 偏对数正态级联分布。

Relationship to fat-tailed distributions

A fat-tailed distribution is a distribution for which the probability density function, for large x, goes to zero as a power [math]\displaystyle{ 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 模板:Contradict-inline. Many other heavy-tailed distributions such as the log-logistic and 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]\displaystyle{ 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模板:Definition

There are parametric (see Embrechts et al.) and non-parametric (see, e.g., Novak) approaches to the problem of the tail-index estimation.

对于尾指数估计的问题，有参数方法（参见Emprechts等人）和非参数方法（例如，Novak）两种。

To estimate the tail-index using the parametric approach, some authors employ GEV distribution or Pareto distribution; they may apply the maximum-likelihood estimator (MLE).

为了使用参数化方法估计尾指数，有些作者采用了GEV分布或帕累托分布；他们可能会运用最大似然估计器（MLE）。

Pickand's tail-index estimator Pickand的尾指数估算器

With [math]\displaystyle{ (X_n , n \geq 1) }[/math] a random sequence of independent and same density function [math]\displaystyle{ F \in D(H(\xi)) }[/math], the Maximum Attraction Domain of the generalized extreme value density [math]\displaystyle{ H }[/math], where [math]\displaystyle{ \xi \in \mathbb{R} }[/math]. If [math]\displaystyle{ \lim_{n\to\infty} k(n) = \infty }[/math] and [math]\displaystyle{ \lim_{n\to\infty} \frac{k(n)}{n}= 0 }[/math], then the Pickands tail-index estimation is

对于[math]\displaystyle{ (X_n , n \geq 1) }[/math]的独立且相同密度函数[math]\displaystyle{ F \in D(H(\xi)) }[/math]的随机序列，广义极值密度[math]\displaystyle{ H }[/math]的最大吸引域，其中[math]\displaystyle{ \xi \in \mathbb{R} }[/math]。如果[math]\displaystyle{ \lim_{n\to\infty} k(n) = \infty }[/math]和[math]\displaystyle{ \lim_{n\to\infty} \frac{k(n)}{n}= 0 }[/math]，则Pickands尾部指数估计为

[math]\displaystyle{ \xi^\text{Pickands}_{(k(n),n)} =\frac{1}{\ln 2} \ln \left( \frac{X_{(n-k(n)+1,n)} - X_{(n-2k(n)+1,n)}}{X_{(n-2k(n)+1,n)} - X_{(n-4k(n)+1,n)}}\right) }[/math]

where [math]\displaystyle{ X_{(n-k(n)+1,n)}=\max \left(X_{n-k(n)+1},\ldots ,X_{n}\right) }[/math]. This estimator converges in probability to [math]\displaystyle{ \xi }[/math].

其中[math]\displaystyle{ X _ {（n-k（n）+ 1，n）} = \ max \ left（X_ {n-k（n）+1}，\ ldots，X_ {n} \ right）\lt / math\gt 。 此估计量的概率收敛到\lt math\gt \ xi \lt / math\gt 。 === Hill's tail-index estimator === Let \lt math\gt (X_t , t \geq 1) }[/math] be a sequence of independent and identically distributed random variables with distribution function [math]\displaystyle{ F \in D(H(\xi)) }[/math], the maximum domain of attraction of the generalized extreme value distribution [math]\displaystyle{ H }[/math], where [math]\displaystyle{ \xi \in \mathbb{R} }[/math]. The sample path is [math]\displaystyle{ {X_t: 1 \leq t \leq n} }[/math] where [math]\displaystyle{ n }[/math] is the sample size. If [math]\displaystyle{ \{k(n)\} }[/math] is an intermediate order sequence, i.e. [math]\displaystyle{ k(n) \in \{1,\ldots,n-1\}, }[/math], [math]\displaystyle{ k(n) \to \infty }[/math] and [math]\displaystyle{ k(n)/n \to 0 }[/math], then the Hill tail-index estimator is

[math]\displaystyle{ \xi^\text{Hill}_{(k(n),n)} = \left(\frac 1 {k(n)} \sum_{i=n-k(n)+1}^n \ln(X_{(i,n)}) - \ln (X_{(n-k(n)+1,n)})\right)^{-1}, }[/math]

where [math]\displaystyle{ X_{(i,n)} }[/math] is the [math]\displaystyle{ i }[/math]-th order statistic of [math]\displaystyle{ X_1, \dots, X_n }[/math]. This estimator converges in probability to [math]\displaystyle{ \xi }[/math], and is asymptotically normal provided [math]\displaystyle{ k(n) \to \infty }[/math] is restricted based on a higher order regular variation property . Consistency and asymptotic normality extend to a large class of dependent and heterogeneous sequences, irrespective of whether [math]\displaystyle{ X_t }[/math] is observed, or a computed residual or filtered data from a large class of models and estimators, including mis-specified models and models with errors that are dependent.

Ratio estimator of the tail-index

The ratio estimator (RE-estimator) of the tail-index was introduced by Goldie and Smith. It is constructed similarly to Hill's estimator but uses a non-random "tuning parameter".

A comparison of Hill-type and RE-type estimators can be found in Novak.

Software

• aest, C tool for estimating the heavy-tail index.

Estimation of heavy-tailed density

Nonparametric approaches to estimate heavy- and superheavy-tailed probability density functions were given in Markovich. These are approaches based on variable bandwidth and long-tailed kernel estimators; on the preliminary data transform to a new random variable at finite or infinite intervals which is more convenient for the estimation and then inverse transform of the obtained density estimate; and "piecing-together approach" which provides a certain parametric model for the tail of the density and a non-parametric model to approximate the mode of the density. Nonparametric estimators require an appropriate selection of tuning (smoothing) parameters like a bandwidth of kernel estimators and the bin width of the histogram. The well known data-driven methods of such selection are a cross-validation and its modifications, methods based on the minimization of the mean squared error (MSE) and its asymptotic and their upper bounds. A discrepancy method which uses well-known nonparametric statistics like Kolmogorov-Smirnov's, von Mises and Anderson-Darling's ones as a metric in the space of distribution functions (dfs) and quantiles of the later statistics as a known uncertainty or a discrepancy value can be found in. Bootstrap is another tool to find smoothing parameters using approximations of unknown MSE by different schemes of re-samples selection, see e.g.

See also

• Leptokurtic distribution
• Generalized extreme value distribution
• Outlier
• Long tail
• Power law
• Seven states of randomness
• Fat-tailed distribution
  • Taleb distribution and Holy grail distribution

References

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