重尾分布

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此词条Jie翻译。已由Smile审校。


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。


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 。



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。）


定义

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[math]\displaystyle{ X }[/math]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]或突变理论 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 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. 


Category:Tails of probability distributions

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Category:Types of probability distributions

类别: 概率分布的类型

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Category:Risk

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