重尾分布

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In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded:[1] 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 Heavy-tailed distributions是指其尾部呈现出不受指数限制的概率分布[1]：也就是说，它们的尾部比指数分布 exponential distribution “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：胖尾分布 Fat-tailed distribution，长尾分布 Long-tailed distribution和次指数分布 Subexponential distributions。实际上，所有常用的重尾分布都属于次指数分布类 subexponential class 。


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” Heavy-tailed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对数正态分布 long-normal distributions ，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

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, MX(t), is infinite for all t > 0.[2]

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, MX(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）。[2]


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

也可以写成尾分布函数 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，则称具有分布函数F的随机变量X的分布为有较长的右尾，

[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具有直观的解释，即如果长尾量超过某个高水平，则概率将接近1，它将超过其他更高的水平。


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，但反过来不一定成立，且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 Convolution 定义的。对于具有共同分布函数[math]\displaystyle{ F }[/math]的两个独立且分布均匀的随机变量[math]\displaystyle{ X_1,X_2 }[/math]，[math]\displaystyle{ F }[/math]与自身的卷积，[math]\displaystyle{ F^{*2} }[/math]是卷积的平方，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[1][3][4] 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[1][5][6]


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


This implies[7] that, for any [math]\displaystyle{ n \geq 1 }[/math],

这意味着[7]，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[7] 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],

对此的概率解释[7]是，对于具有共同分布[math]\displaystyle{ F }[/math]的[math]\displaystyle{ n }[/math]个独立随机变量[math]\displaystyle{ X_1,\ldots,X_n }[/math]的总和


[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[8] or catastrophe principle.[9]

这通常被称为单跳 single big jump[10]或浩劫原则 catastrophe principle [11]。


A distribution [math]\displaystyle{ F }[/math] on the whole real line is subexponential if the distribution [math]\displaystyle{ F I([0,\infty)) }[/math] is.[12] 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))\lt /m4ath\gt 为实数，则\lt math\gt F }[/math]为整个实数上的次指数分布。[13]此时[math]\displaystyle{ I([0,\infty)) }[/math]是正半轴的指标函数。或者，当且仅当[math]\displaystyle{ X^+ = \max(0,X) }[/math]是次指数时，实数上支持的随机变量[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，但可以构造出非次指数分布的长尾分布的示例。

Common heavy-tailed distributions 常见的重尾分布

All commonly used heavy-tailed distributions are subexponential.[7]

Those that are one-tailed include:

Those that are two-tailed include:


All commonly used heavy-tailed distributions are subexponential.[6]

所有常用的重尾分布都是次指数的。[7]

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。[18][19]

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[20]，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（或有半线的支持），例如莱维分布。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 The skew lognormal cascade distribution。[21]



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]\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 [22]. 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，对于较大的x，概率密度函数为[math]\displaystyle{ x^{-a} }[/math]趋于零。由于这样的幂总是受到指数分布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[23]。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 尾指数估计

There are parametric (see Embrechts et al.[7]) and non-parametric (see, e.g., Novak[24]) 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[7]）和非参数方法（例如，Novak[24]）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；他们可能会运用极大似然估计方法（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[25] 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[7][25]

对于[math]\displaystyle{ (X_n , n \geq 1) }[/math]的独立且相同的密度函数[math]\displaystyle{ F \in D(H(\xi)) }[/math]的随机序列，是广义极值密度 the generalized extreme value density [math]\displaystyle{ H }[/math]的最大吸引域 the Maximum Attraction Domain [25]，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[7][25]


[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) }[/math]。 此估计量的概率收敛到[math]\displaystyle{ \xi }[/math]。


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

Let [math]\displaystyle{ (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[26]

令[math]\displaystyle{ (X_t , t \geq 1) }[/math]为具有分布函数[math]\displaystyle{ F \in D(H(\xi)) }[/math]独立且均匀分布的随机变量序列，其分布函数为广义极值分布[math]\displaystyle{ H }[/math]的最大吸引域，其中[math]\displaystyle{ \xi \in \mathbb{R} }[/math]。样本路径为[math]\displaystyle{ {X_t: 1 \leq t \leq n} }[/math]，其中[math]\displaystyle{ n }[/math]为样本大小。 如果[math]\displaystyle{ \{k(n)\} }[/math]是中间阶数序列，即[math]\displaystyle{ k(n) \in \{1,\ldots,n-1\}, }[/math]，[math]\displaystyle{ k(n) \to \infty }[/math]和[math]\displaystyle{ k(n)/n \to 0 }[/math]，则Hill尾指数估计器为[27]：


[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[28] .[29] Consistency and asymptotic normality extend to a large class of dependent and heterogeneous sequences,[30][31] 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.[32][33][34]

其中[math]\displaystyle{ X_{(i,n)} }[/math]是[math]\displaystyle{ X_1, \dots, X_n }[/math]的第[math]\displaystyle{ i }[/math]次序统计量。该估计量依概率收敛于[math]\displaystyle{ \xi }[/math]，并且在基于高阶的正则变化性质的情况下，是限制[math]\displaystyle{ k(n) \to \infty }[/math]的渐近正态[35].[36]。一致性和渐近正态性适用于一大类相关序列和异类序列[37][38]，而不管是否观测到[math]\displaystyle{ X_t }[/math]，或者来自大量模型和估计量（包括错误指定的模型和具有相关误差的模型）计算出的残差或筛选数据。[39][40][41]


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 and Smith.[42] It is constructed similarly to Hill's estimator but uses a non-random "tuning parameter".

尾指数的比率估计器（RE估计器）由Goldie和Smith提出[43]。它的构造类似于Hill估计器，但使用了非随机的“调整参数”


A comparison of Hill-type and RE-type estimators can be found in Novak.[24]

在Novak中可以找到Hill型和RE型估计量的比较。[24]

Software 应用软件

  • aest, C tool for estimating the heavy-tail index.[44]
  • 用于估计重尾指数的软件aest和C。[45]

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 Markovich.[46] 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.[47] 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.[46] Bootstrap is another tool to find smoothing parameters using approximations of unknown MSE by different schemes of re-samples selection, see e.g.[48]

Markovich中给出了估计重尾和超重尾概率密度函数的非参数方法。[46]这些是基于可变带宽 variable bandwidth和长尾核估计器 long-tailed kernel estimators的方法。将初步数据以有限或无限间隔变换为新的随机变量，这样更便于估计，然后对获得的密度估计进行逆变换；以及“拼合方法”，它为密度的尾部提供了确定的参数模型，并为近似密度模型提供了非参数模型。非参数估计器需要适当选择调整（平滑）参数，例如内核估计器的带宽和直方图的组距。这种选择大众化数据驱动方法是基于均方误差（MSE）及其渐近或上限的最小化的交叉验证及修改方法。[47]可以找到一种差异方法，通过使用著名的非参数统计数据（例如Kolmogorov-Smirnov's，von Mises和Anderson-Darling的统计量）作为分布函数（dfs）空间中的度量，并将后来的统计量的分位数作为已知的不确定性或差异值。[46]自助法 Bootstrap是另一种工具，可以通过不同的重抽样方案使用未知MSE的近似值来查找平滑参数。[48]


See also 其他参考资料


  • 尖峭态分布 Leptokurtic distribution
  • 广义极值分布 Generalized extreme value distribution
  • 离群值 Outlier
  • 长尾 Long tail
  • 幂律 Power law
  • 随机的七个状态 Seven states of randomness
  • 胖尾分布 Fat-tailed distribution
    • 塔勒布分布 Taleb distribution和圣杯分布 Holy grail distribution


References 参考文献

  1. ↑ 1.0 1.1 1.2 1.3 Asmussen, S. R. (2003). "Steady-State Properties of GI/G/1". Applied Probability and Queues. Stochastic Modelling and Applied Probability. 51. pp. 266–301. doi:10.1007/0-387-21525-5_10. ISBN 978-0-387-00211-8. 
  2. ↑ 2.0 2.1 Rolski, Schmidli, Scmidt, Teugels, Stochastic Processes for Insurance and Finance, 1999
  3. ↑ Chistyakov, V. P. (1964). "A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes". ResearchGate (in English). Retrieved April 7, 2019.
  4. ↑ Teugels, Jozef L. (1975). "The Class of Subexponential Distributions". University of Louvain: Annals of Probability. Retrieved April 7, 2019.
  5. ↑ Chistyakov, V. P. (1964). "A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes". ResearchGate (in English). Retrieved April 7, 2019.
  6. ↑ Teugels, Jozef L. (1975). "The Class of Subexponential Distributions". University of Louvain: Annals of Probability. Retrieved April 7, 2019.
  7. ↑ 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 Embrechts P.; Klueppelberg C.; Mikosch T. (1997). Modelling extremal events for insurance and finance. Stochastic Modelling and Applied Probability. 33. Berlin: Springer. doi:10.1007/978-3-642-33483-2. ISBN 978-3-642-08242-9. 
  8. ↑ Foss, S.; Konstantopoulos, T.; Zachary, S. (2007). "Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments" (PDF). Journal of Theoretical Probability. 20 (3): 581. arXiv:math/0509605. CiteSeerX 10.1.1.210.1699. doi:10.1007/s10959-007-0081-2.
  9. ↑ Wierman, Adam (January 9, 2014). "Catastrophes, Conspiracies, and Subexponential Distributions (Part III)". Rigor + Relevance blog. RSRG, Caltech. Retrieved January 9, 2014.
  10. ↑ Foss, S.; Konstantopoulos, T.; Zachary, S. (2007). "Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments" (PDF). Journal of Theoretical Probability. 20 (3): 581. arXiv:math/0509605. CiteSeerX 10.1.1.210.1699. doi:10.1007/s10959-007-0081-2.
  11. ↑ Wierman, Adam (January 9, 2014). "Catastrophes, Conspiracies, and Subexponential Distributions (Part III)". Rigor + Relevance blog. RSRG, Caltech. Retrieved January 9, 2014.
  12. ↑ Willekens, E. (1986). "Subexponentiality on the real line". Technical Report. K.U. Leuven.
  13. ↑ Willekens, E. (1986). "Subexponentiality on the real line". Technical Report. K.U. Leuven.
  14. ↑ Falk, M., Hüsler, J. & Reiss, R. (2010). Laws of Small Numbers: Extremes and Rare Events. Springer. p. 80. ISBN 978-3-0348-0008-2. 
  15. ↑ Alves, M.I.F., de Haan, L. & Neves, C. (March 10, 2006). "Statistical inference for heavy and super-heavy tailed distributions" (PDF). Archived from the original (PDF) on June 23, 2007. Retrieved November 1, 2011.{{cite web}}: CS1 maint: multiple names: authors list (link)
  16. ↑ John P. Nolan (2009). "Stable Distributions: Models for Heavy Tailed Data" (PDF). Retrieved 2009-02-21.
  17. ↑ Stephen Lihn (2009). "Skew Lognormal Cascade Distribution". Archived from the original on 2014-04-07. Retrieved 2009-06-12.
  18. ↑ Falk, M., Hüsler, J. & Reiss, R. (2010). Laws of Small Numbers: Extremes and Rare Events. Springer. p. 80. ISBN 978-3-0348-0008-2. 
  19. ↑ Alves, M.I.F., de Haan, L. & Neves, C. (March 10, 2006). "Statistical inference for heavy and super-heavy tailed distributions" (PDF). Archived from the original (PDF) on June 23, 2007. Retrieved November 1, 2011.{{cite web}}: CS1 maint: multiple names: authors list (link)
  20. ↑ John P. Nolan (2009). "Stable Distributions: Models for Heavy Tailed Data" (PDF). Retrieved 2009-02-21.
  21. ↑ Stephen Lihn (2009). "Skew Lognormal Cascade Distribution". Archived from the original on 2014-04-07. Retrieved 2009-06-12.
  22. ↑ 模板:Contradict-inline
  23. ↑ 模板:Contradict-inline
  24. ↑ 24.0 24.1 24.2 24.3 Novak S.Y. (2011). Extreme value methods with applications to finance. London: CRC. ISBN 978-1-43983-574-6. 
  25. ↑ 25.0 25.1 25.2 25.3 Pickands III, James (Jan 1975). "Statistical Inference Using Extreme Order Statistics". The Annals of Statistics. 3 (1): 119–131. doi:10.1214/aos/1176343003. JSTOR 2958083.
  26. ↑ Hill B.M. (1975) A simple general approach to inference about the tail of a distribution. Ann. Stat., v. 3, 1163–1174.
  27. ↑ Hill B.M. (1975) A simple general approach to inference about the tail of a distribution. Ann. Stat., v. 3, 1163–1174.
  28. ↑ Hall, P.(1982) On some estimates of an exponent of regular variation. J. R. Stat. Soc. Ser. B., v. 44, 37–42.
  29. ↑ Haeusler, E. and J. L. Teugels (1985) On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann. Stat., v. 13, 743–756.
  30. ↑ Hsing, T. (1991) On tail index estimation using dependent data. Ann. Stat., v. 19, 1547–1569.
  31. ↑ Hill, J. (2010) On tail index estimation for dependent, heterogeneous data. Econometric Th., v. 26, 1398–1436.
  32. ↑ Resnick, S. and Starica, C. (1997). Asymptotic behavior of Hill’s estimator for autoregressive data. Comm. Statist. Stochastic Models 13, 703–721.
  33. ↑ Ling, S. and Peng, L. (2004). Hill’s estimator for the tail index of an ARMA model. J. Statist. Plann. Inference 123, 279–293.
  34. ↑ Hill, J. B. (2015). Tail index estimation for a filtered dependent time series. Stat. Sin. 25, 609–630.
  35. ↑ Hall, P.(1982) On some estimates of an exponent of regular variation. J. R. Stat. Soc. Ser. B., v. 44, 37–42.
  36. ↑ Haeusler, E. and J. L. Teugels (1985) On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann. Stat., v. 13, 743–756.
  37. ↑ Hsing, T. (1991) On tail index estimation using dependent data. Ann. Stat., v. 19, 1547–1569.
  38. ↑ Hill, J. (2010) On tail index estimation for dependent, heterogeneous data. Econometric Th., v. 26, 1398–1436.
  39. ↑ Resnick, S. and Starica, C. (1997). Asymptotic behavior of Hill’s estimator for autoregressive data. Comm. Statist. Stochastic Models 13, 703–721.
  40. ↑ Ling, S. and Peng, L. (2004). Hill’s estimator for the tail index of an ARMA model. J. Statist. Plann. Inference 123, 279–293.
  41. ↑ Hill, J. B. (2015). Tail index estimation for a filtered dependent time series. Stat. Sin. 25, 609–630.
  42. ↑ Goldie C.M., Smith R.L. (1987) Slow variation with remainder: theory and applications. Quart. J. Math. Oxford, v. 38, 45–71.
  43. ↑ Goldie C.M., Smith R.L. (1987) Slow variation with remainder: theory and applications. Quart. J. Math. Oxford, v. 38, 45–71.
  44. ↑ Crovella, M. E.; Taqqu, M. S. (1999). "Estimating the Heavy Tail Index from Scaling Properties". Methodology and Computing in Applied Probability. 1: 55–79. doi:10.1023/A:1010012224103.
  45. ↑ Crovella, M. E.; Taqqu, M. S. (1999). "Estimating the Heavy Tail Index from Scaling Properties". Methodology and Computing in Applied Probability. 1: 55–79. doi:10.1023/A:1010012224103.
  46. ↑ 46.0 46.1 46.2 46.3 Markovich N.M. (2007). Nonparametric Analysis of Univariate Heavy-Tailed data: Research and Practice. Chitester: Wiley. ISBN 978-0-470-72359-3. 
  47. ↑ 47.0 47.1 Wand M.P., Jones M.C. (1995). Kernel smoothing. New York: Chapman and Hall. ISBN 978-0412552700. 
  48. ↑ 48.0 48.1 Hall P. (1992). The Bootstrap and Edgeworth Expansion. Springer. ISBN 9780387945088. 


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