Subexponential distributions
Subexponential distributions
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DOI：
10.1002/9780470012505.tas039
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发表时间：
1998-08
期刊：
影响因子：
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通讯作者：
C. Goldie;C. Klüppelberg
中科院分区：
文献类型：
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作者：
C. Goldie;C. Klüppelberg
我们调查了亚指数概率分布类的属性和用途，特别关注它们在建模重尾数据（例如保险和排队应用中出现的数据）中的用途。我们给出了一个详细的总结的核心理论，并讨论了次指数在各种情况下，包括极端，随机游动和Levy过程负漂移，和随机变量的总和，后者扩展到涵盖随机和，加权和和移动平均。1.次指数分布是一类特殊的重尾分布。这个名字来源于它们的一个性质，即它们的尾部比任何指数尾部下降得更慢，见式（1 - 4）。这意味着大值可能以不可忽略的概率出现在样本中，并且使得次指数分布成为建模情况的候选者，其中与数据的平均大小相比，样本中出现了一些非常大的值。这种模式经常出现在保险数据中，例如在风、风暴或洪水保险（统称为巨灾保险）中。次指数索赔可以解释公司盈余过程中的大幅波动，增加了此类投资组合的风险。这种情况将在第2节中讨论。次指数在模型中也扮演着类似的角色。在服务时间极端的情况下，用次指数分布建模，会导致系统中的等待时间很长（见例2.7）。工作负荷过程也显示出很大的波动（见例6.4）。线性模型被广泛用作（或一阶近似）相关数据的简单模型。创新中的极大值，由次指数分布建模，对单次观察有直接影响。此外，它们在样品的较大部分中引起效应，由线性滤波器确定。在所有这些模型中，一些大的值可以确定系统的长期行为。这可以非常精确地通过描述的样本路径行为的随机过程的剩余过程中的保险或工作量过程的队列，因为后者的模型已经得到了最充分的研究。第6节对此进行了审查。重尾只是次指数分布的deening性质的结果之一，其被专门设计为与在上述应用领域中通常采用的概率模型良好地工作。次指数概念的一般性水平正好为1
We survey the properties and uses of the class of subexponential probability distributions, paying particular attention to their use in modelling heavy-tailed data such as occurs in insurance and queueing applications. We give a detailed summary of the core theory and discuss subexponen-tiality in various contexts including extremes, random walks and L evy processes with negative drift, and sums of random variables, the latter extended to cover random sums, weighted sums and moving averages. 1. Deenition and rst properties Subexponential distributions are a special class of heavy{tailed distributions. The name arises from one of their properties, that their tails decrease more slowly than any exponential tail; see (1.4). This implies that large values can occur in a sample with non{negligible probability, and makes the subexponen-tial distributions candidates for modelling situations where some extremely large values occur in a sample compared to the mean size of the data. Such a pattern is often seen in insurance data, for instance in re, wind{storm or ood insurance (collectively known as catastrophe insurance). Subexponential claims can account for large uctuations in the surplus process of a company, increasing the risk involved in such portfolios. This situation is treated in Section 2. Subexponentials play a similar role in queueing models. Situations with extreme service times, modelled by a subexponential distribution, result in huge waiting times in the system (see Example 2.7). The workload process also shows large uctuations (see Example 6.4). Linear models are widely used as simple models for (or rst order approximations to) dependent data. Extremely large values in the innovations, modelled by subexponential distributions, have immediate consequences for the single observation. Moreover, they cause eeects in larger parts of the sample, determined by the linear lter. In all these models a few large values may determine the long{term behaviour of a system. This can be made very precise by describing the sample path behaviour of resulting stochastic processes as the surplus process in insurance or the workload process of a queue, since the latter models have been the most fully investigated. This is reviewed in Section 6. Heavy tails are just one of the consequences of the deening property of subexponential distributions, which is designed specially to work well with the probabilistic models commonly employed in the above{mentioned areas of application. The subexponential concept has just the right level of generality to 1